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Spin covers of maximal compact subgroups of Kac–Moody groups and spin-extended Weyl groups

David Ghatei
• School of Mathematics, University of Birmingham, Edgbaston, Birmingham, B15 2TT, United Kingdom
• Other articles by this author:
/ Max Horn
/ Ralf Köhl
/ Sebastian Weiß
Published Online: 2016-08-09 | DOI: https://doi.org/10.1515/jgth-2016-0034

Abstract

Let G be a split real Kac–Moody group of arbitrary type and let K be its maximal compact subgroup, i.e. the subgroup of elements fixed by a Cartan–Chevalley involution of G. We construct non-trivial spin covers of K, thus confirming a conjecture by Damour and Hillmann. For irreducible simply-laced diagrams and for all spherical diagrams these spin covers are two-fold central extensions of K. For more complicated irreducible diagrams these spin covers are central extensions by a finite 2-group of possibly larger cardinality. Our construction is amalgam-theoretic and makes use of the generalised spin representations of maximal compact subalgebras of split real Kac–Moody algebras studied by Hainke, Levy and the third author. Our spin covers contain what we call spin-extended Weyl groups which admit a presentation by generators and relations obtained from the one for extended Weyl groups by relaxing the condition on the generators so that only their eighth powers are required to be trivial.

1 Introduction

In [9, Section 3.5] it turned out that the existence of a spin-extended Weyl group ${W}^{\mathrm{spin}}\left({E}_{10}\right)$ would be very useful for the study of fermionic billards. Lacking a concrete mathematical model of that group ${W}^{\mathrm{spin}}\left({E}_{10}\right)$, Damour and Hillmann in their article instead use images of ${W}^{\mathrm{spin}}\left({E}_{10}\right)$ afforded by various generalised spin representations as described in [10, 11], which can be realised as matrix groups.

In [9, Section 3.5, footnote 18, p. 24], Damour and Hillmann conjecture that the spin-extended Weyl group ${W}^{\mathrm{spin}}\left({E}_{10}\right)$ can be constructed as a discrete subgroup of a double spin cover $\mathrm{Spin}\left({E}_{10}\right)$ of the subgroup $K\left({E}_{10}\right)$ of elements fixed by the Cartan–Chevalley involution of the split real Kac–Moody group of type ${E}_{10}$. The purpose of this article is to confirm this conjecture, and to generalise it to arbitrary diagrams resp. arbitrary generalised Cartan matrices

In the simply-laced case our result is as follows.

Let Π be an irreducible simply-laced Dynkin diagram, i.e., a Dynkin diagram affording only single edges, let $I\mathrm{=}\mathrm{\left\{}\mathrm{1}\mathrm{,}\mathrm{\dots }\mathrm{,}n\mathrm{\right\}}$ be a set of labels of the vertices of Π, and let $K\mathit{}\mathrm{\left(}\mathrm{\Pi }\mathrm{\right)}$ be the subgroup of elements fixed by the Cartan–Chevalley involution of the split real Kac–Moody group of type Π. For each $i\mathrm{\in }I$ let ${G}_{i}\mathrm{\cong }\mathrm{Spin}\mathit{}\mathrm{\left(}\mathrm{2}\mathrm{\right)}$ and for each $i\mathrm{\ne }j\mathrm{\in }I$ let

${G}_{ij}\cong \left\{\begin{array}{cc}\mathrm{Spin}\left(3\right),\hfill & \mathit{\text{if}}i\mathit{\text{,}}j\mathit{\text{form an edge of}}\mathrm{\Pi }\mathit{\text{,}}\hfill \\ \left(\mathrm{Spin}\left(2\right)×\mathrm{Spin}\left(2\right)\right)/〈\left(-1,-1\right)〉,\hfill & \mathit{\text{if}}i\mathit{\text{,}}j\mathit{\text{do not form an edge of}}\mathrm{\Pi }\mathit{\text{.}}\hfill \end{array}$

Moreover, for $i\mathrm{,}j\mathrm{\in }I$ with $i\mathrm{<}j$, let ${\varphi }_{i\mathit{}j}^{i}\mathrm{:}{G}_{i}\mathrm{\to }{G}_{i\mathit{}j}$ be the standard embedding as “upper-left diagonal block” and let ${\varphi }_{i\mathit{}j}^{j}\mathrm{:}{G}_{j}\mathrm{\to }{G}_{i\mathit{}j}$ be the standard embedding as “lower-right diagonal block”.

Then up to isomorphism there exists a uniquely determined group, denoted $\mathrm{Spin}\mathit{}\mathrm{\left(}\mathrm{\Pi }\mathrm{\right)}$, whose multiplication table extends the partial multiplication provided by $\mathrm{\left(}{\mathrm{\bigsqcup }}_{i\mathrm{<}j\mathrm{\in }I}{G}_{i\mathit{}j}\mathrm{\right)}\mathrm{/}\mathrm{\sim }$, where $\mathrm{\sim }$ is the equivalence relation determined by

${\varphi }_{ij}^{i}\left(x\right)\sim {\varphi }_{ik}^{i}\left(x\right)$

for all $i\mathrm{\ne }j\mathrm{,}k\mathrm{\in }I$ and $x\mathrm{\in }{G}_{i}$. Furthermore, there exists a canonical two-to-one central extension $\mathrm{Spin}\mathit{}\mathrm{\left(}\mathrm{\Pi }\mathrm{\right)}\mathrm{\to }K\mathit{}\mathrm{\left(}\mathrm{\Pi }\mathrm{\right)}$.

The system $\left\{{G}_{i},{G}_{ij},{\varphi }_{ij}^{i}\right\}$ is called an amalgam of groups, the pair consisting of the group $\mathrm{Spin}\left(\mathrm{\Pi }\right)$ and the set of canonical embeddings ${\tau }_{i}:{G}_{i}↪\mathrm{Spin}\left(\mathrm{\Pi }\right)$, ${\tau }_{ij}:{G}_{ij}↪\mathrm{Spin}\left(\mathrm{\Pi }\right)$ a universal enveloping group; the canonical embeddings are called enveloping homomorphisms. Formal definitions and background information concerning amalgams can be found in Section 3. Since all ${G}_{i}\cong \mathrm{Spin}\left(2\right)$ are isomorphic to one another, it in fact suffices to fix one group $U\cong \mathrm{Spin}\left(2\right)$ instead with connecting homomorphisms ${\varphi }_{ij}^{i}:U\to {G}_{ij}$.

The formalization of the concept of standard embedding as “upper-left/lower right diagonal block” can be found in Section 10. Note that, since the ${G}_{i}$ are only given up to isomorphism, these standard embeddings are only well-defined up to automorphism of ${G}_{i}$, which leads to some ambiguity. Since by [21] the group $K\left(\mathrm{\Pi }\right)$ (and therefore each of its central extensions by a finite group) is a topological group, one may assume the ${\varphi }_{ij}^{i}$ to be continuous, thus restricting oneself to the ambiguity stemming from the two continuous automorphisms of $\mathrm{Spin}\left(2\right)$, the identity and the inversion homomorphisms. This ambiguity is resolved in Section 10.

Theorem A provides us with the means of characterizing ${W}^{\mathrm{spin}}\left(\mathrm{\Pi }\right)$.

Let $\mathrm{\Pi }$ be an irreducible simply-laced Dynkin diagram, $I\mathrm{=}\mathrm{\left\{}\mathrm{1}\mathrm{,}\mathrm{\dots }\mathrm{,}n\mathrm{\right\}}$ a set of labels of the vertices of Π, and for each $i\mathrm{\in }I$ let

• ${\tau }_{i}:{G}_{i}\cong \mathrm{Spin}\left(2\right)↪\mathrm{Spin}\left(\mathrm{\Pi }\right)$ be the canonical enveloping homomorphisms,

• ${x}_{i}\in {G}_{i}$ be elements of order eight whose polar coordinates involve the angle $\frac{\pi }{4}$,

• ${r}_{i}:={\tau }_{i}\left({x}_{i}\right)$.

Then ${W}^{\mathrm{spin}}\mathit{}\mathrm{\left(}\mathrm{\Pi }\mathrm{\right)}\mathrm{:=}\mathrm{〈}{r}_{i}\mathrm{\mid }i\mathrm{\in }I\mathrm{〉}$ satisfies the defining relations

${r}_{i}^{8}=1,$(R1)${r}_{i}^{-1}{r}_{j}^{2}{r}_{i}={r}_{j}^{2}{r}_{i}^{2n\left(i,j\right)}$$\mathrm{ }\mathit{\text{for}}i\ne j\in I,$(R2)$\underset{{m}_{ij}\mathit{\text{factors}}}{\underset{⏟}{{r}_{i}{r}_{j}{r}_{i}\mathrm{\cdots }}}=\underset{{m}_{ij}\mathit{\text{factors}}}{\underset{⏟}{{r}_{j}{r}_{i}{r}_{j}\mathrm{\cdots }}}$$\mathrm{ }\mathit{\text{for}}i\ne j\in I,$(R3)

where

${m}_{ij}=\left\{\begin{array}{cc}3,\hfill & \mathit{\text{if}}i\mathit{\text{,}}j\mathit{\text{form an edge,}}\hfill \\ 2,\hfill & \mathit{\text{if}}i\mathit{\text{,}}j\mathit{\text{do not form an edge,}}\hfill \end{array}$

and

$n\left(i,j\right)=\left\{\begin{array}{cc}1,\hfill & \mathit{\text{if}}i\mathit{\text{,}}j\mathit{\text{form an edge,}}\hfill \\ 0,\hfill & \mathit{\text{if}}i\mathit{\text{,}}j\mathit{\text{do not form an edge.}}\hfill \end{array}$

To be a set of defining relations means that any product of the ${r}_{i}$ that in ${W}^{\mathrm{spin}}\left(\mathrm{\Pi }\right)$ represents the identity can be written as a product of conjugates of ways of representing the identity via (R1), (R2), (R3).

Our results in fact can be extended to arbitrary diagrams as discussed in Sections 16, 17, and 18.

As a by-product of our proof of Theorem A we show in Section 19 that for non-spherical diagrams Π the groups $\mathrm{Spin}\left(\mathrm{\Pi }\right)$ and $K\left(\mathrm{\Pi }\right)$ are never simple; instead they always admit a non-trivial compact Lie group as a quotient via the generalised spin representation described in [20]. The generalised spin representation of $\mathrm{Spin}\left(\mathrm{\Pi }\right)$ is continuous, so that the obtained normal subgroups are closed. Similar non-simplicity phenomena as abstract groups have been observed in [5]. Furthermore, we observe that for arbitrary simply-laced diagrams the image of ${W}^{\mathrm{spin}}$ under the generalised spin representation is finite, generalizing [9, Lemma 2, p. 49].

Sections 3, 4, 5, 6 and 8 are introductory in nature; we revise the notions of amalgams, Cartan matrices and Dynkin diagrams and fix our notation for orthogonal and spin groups. Sections 9 and 10 deal with the classification theory of amalgams and, as a blueprint for Theorem A, identify $\mathrm{SO}\left(n\right)$ and $\mathrm{Spin}\left(n\right)$ as universal enveloping groups of $\mathrm{SO}\left(2\right)$-, resp. $\mathrm{Spin}\left(2\right)$-amalgams of type ${A}_{n-1}$. In Section 11 we prove Theorem A.

Sections 13, 14 and 15 provide us with the necessary tools for generalizing our findings to arbitrary diagrams; they deal with equivariant coverings of the real projective plane by the split Cayley hexagon and the symplectic quadrangle and with coverings of the real projective plane and the symplectic quadrangle by trees. In Section 16 we study $\mathrm{SO}\left(2\right)$- and $\mathrm{Spin}\left(2\right)$-amalgams for this larger class of diagrams. Section 17 deals with the general version of Theorem A. Section 18 deals with the proof of Theorem B and its generalization. In Section 19 we observe that our findings provide epimorphisms from $\mathrm{Spin}\left(\mathrm{\Pi }\right)$ and $K\left(\mathrm{\Pi }\right)$ onto non-trivial compact Lie groups.

2 Conventions

$ℕ:=\left\{1,2,3,\mathrm{\dots }\right\}$ denotes the set of positive integers.

Throughout this article we use the convention $ij:=\left\{i,j\right\}$ if the set $\left\{i,j\right\}$ is used as an index. For example, if ${G}_{ij}$ is a group, then ${G}_{ji}$ is the same group. Note that this does not apply to superscripts, so ${G}^{ij}$ and ${G}^{ji}$ may differ.

For any group G, consider the following maps:

$\mathrm{inv}:G\to G,x↦{x}^{-1},$$\mathrm{ }\text{the inverse map},$$\mathrm{sq}:G\to G,x↦{x}^{2},$$\mathrm{ }\text{the square map}.$

Both maps commute with any group homomorphism.

For any group G, we denote by $Z\left(G\right)$ the centre of G.

3 Amalgams

In this section we recall the concept of amalgams. More details concerning this concept can, in various formulations, be found in [2, Part III.$\mathcal{𝒞}$], [25, Section 1.3], [14, Section 1].

Let U be a group, and $I\ne \mathrm{\varnothing }$ a set. A U-amalgam over I is a set

$\mathcal{𝒜}=\left\{{G}_{ij},{\varphi }_{ij}^{i}\mid i\ne j\in I\right\}$

such that ${G}_{ij}$ is a group and ${\varphi }_{ij}^{i}:U\to {G}_{ij}$ is a monomorphism for all $i\ne j\in I$. The maps ${\varphi }_{ij}^{i}$ are called connecting homomorphisms. The amalgam is continuous if U and ${G}_{ij}$ are topological groups, and ${\varphi }_{ij}^{i}$ is continuous for all $i\ne j\in I$.

Let $\stackrel{~}{\mathcal{𝒜}}=\left\{{\stackrel{~}{G}}_{ij},{\stackrel{~}{\varphi }}_{ij}^{i}\mid i\ne j\in I\right\}$ and $\mathcal{𝒜}=\left\{{G}_{ij},{\varphi }_{ij}^{i}\mid i\ne j\in I\right\}$ be U-amalgams over I. An epimorphism, resp. an isomorphism $\alpha :\stackrel{~}{\mathcal{𝒜}}\to \mathcal{𝒜}$ of U-amalgams is a system

$\alpha =\left\{\pi ,{\alpha }_{ij}\mid i\ne j\in I\right\}$

consisting of a permutation $\pi \in Sym\left(I\right)$ and group epimorphisms, resp. isomorphisms

${\alpha }_{ij}:{\stackrel{~}{G}}_{ij}\to {G}_{\pi \left(i\right)\pi \left(j\right)}$

such that for all $i\ne j\in I$

${\alpha }_{ij}\circ {\stackrel{~}{\varphi }}_{ij}^{i}={\varphi }_{\pi \left(i\right)\pi \left(j\right)}^{\pi \left(i\right)},$

that is, the following diagram commutes:

More generally, let

$\stackrel{~}{\mathcal{𝒜}}=\left\{{\stackrel{~}{G}}_{ij},{\stackrel{~}{\varphi }}_{ij}^{i}\mid i\ne j\in I\right\}$

be a U-amalgam and let

$\mathcal{𝒜}=\left\{{G}_{ij},{\varphi }_{ij}^{i}\mid i\ne j\in I\right\}$

be a V-amalgam. An epimorphism $\alpha :\stackrel{~}{\mathcal{𝒜}}\to \mathcal{𝒜}$ is a system

$\alpha =\left\{\pi ,{\rho }^{i},{\alpha }_{ij}\mid i\ne j\in I\right\}$

consisting of a permutation $\pi \in Sym\left(I\right)$, group epimorphisms ${\rho }^{i}:U\to V$ and group epimorphisms

${\alpha }_{ij}:{\stackrel{~}{G}}_{ij}\to {G}_{\pi \left(i\right)\pi \left(j\right)}$

such that for all $i\ne j\in I$,

${\alpha }_{ij}\circ {\stackrel{~}{\varphi }}_{ij}^{i}={\varphi }_{\pi \left(i\right)\pi \left(j\right)}^{\pi \left(i\right)}\circ {\rho }^{\pi \left(i\right)},$

that is, the following diagram commutes:

If (and only if) in the epimorphism $\alpha :\stackrel{~}{\mathcal{𝒜}}\to \mathcal{𝒜}$ each ${\rho }^{i}:U\to V$ is an isomorphism, then one obtains an epimorphism ${\alpha }^{\prime }:\stackrel{~}{\mathcal{𝒜}}\to {\mathcal{𝒜}}^{\prime }$ of U-amalgams by defining ${\alpha }^{\prime }=\left\{\pi ,{\alpha }_{ij}\mid i\ne j\in I\right\}$ and ${\mathcal{𝒜}}^{\prime }=\left\{{G}_{ij},{\left({\varphi }_{ij}^{i}\right)}^{\prime }\right\}$ via

${\left({\varphi }_{ij}^{i}\right)}^{\prime }:U\to {G}_{ij},u↦\left({\varphi }_{ij}^{i}\circ {\rho }^{i}\right)\left(u\right).$

If this ${\alpha }^{\prime }$ turns out to be an isomorphism of U-amalgams, by slight abuse of terminology we also call the epimorphism α an isomorphism of amalgams.

More generally, an amalgam can be defined as a collection of groups ${G}_{i}$ and a collection of groups ${G}_{ij}$ with connecting homomorphisms

${\psi }_{ij}^{i}:{G}_{i}\to {G}_{ij}.$

Since in our situation for all i there exist isomorphisms ${\gamma }_{i}:U\to {G}_{i}$, it suffices to consider the connecting homomorphisms ${\varphi }_{ij}^{i}={\psi }_{ij}^{i}\circ {\gamma }_{i}$.

In the more general setting, an isomorphism of amalgams consists of a permutation π of the index set I and isomorphisms

${\alpha }_{i}:{G}_{i}\to {\overline{G}}_{\pi \left(i\right)}\mathit{ }\text{and}\mathit{ }{\alpha }_{ij}:{G}_{ij}\to {\overline{G}}_{\pi \left(i\right)\pi \left(j\right)}$

such that

${\alpha }_{ij}\circ {\psi }_{ij}^{i}={\overline{\psi }}_{\pi \left(i\right)\pi \left(j\right)}^{\pi \left(i\right)}\circ {\alpha }_{i}.$

A routine calculation shows that U-amalgams and isomorphisms of U-amalgams are special cases of amalgams and isomorphisms of amalgams as found in the literature.

Given a U-amalgam $\mathcal{𝒜}=\left\{{G}_{ij},{\varphi }_{ij}^{i}\mid i\ne j\in I\right\}$, an enveloping group of $\mathcal{𝒜}$ is a pair $\left(G,\tau \right)$ consisting of a group G and a set

$\tau =\left\{{\tau }_{ij}\mid i\ne j\in I\right\}$

of enveloping homomorphisms ${\tau }_{ij}:{G}_{ij}\to G$ such that

$G=〈{\tau }_{ij}\left({G}_{ij}\right)\mid i\ne j\in I〉$

and

${\tau }_{ij}\circ {\varphi }_{ij}^{j}={\tau }_{kj}\circ {\varphi }_{kj}^{j}\mathit{ }\text{for all}i\ne j\ne k\in I,$

that is, for $i\ne j\ne k\in I$ the following diagram commutes:

We write $\tau :\mathcal{𝒜}\to G$ and call τ an enveloping morphism. An enveloping group $\left(G,\tau \right)$ and the corresponding enveloping morphism are faithful if ${\tau }_{ij}$ is a monomorphism for all $i\ne j\in I$.

Given a U-amalgam $\mathcal{𝒜}=\left\{{G}_{ij},{\varphi }_{ij}^{i}\right\}$, an enveloping group $\left(G,\tau \right)$ is called a universal enveloping group if, given an enveloping group $\left(H,{\tau }^{\prime }\right)$ of $\mathcal{𝒜}$, there is a unique epimorphism $\psi :G\to H$ such that for all $i,j\in I$ with $i\ne j$ one has $\psi \circ {\tau }_{ij}={\tau }_{ij}^{\prime }$. We write $\tau :\mathcal{𝒜}\to G$ and call τ a universal enveloping morphism. By universality, two universal enveloping groups $\left({G}_{1},{\tau }_{1}\right)$ and $\left({G}_{2},{\tau }_{2}\right)$ of a U-amalgam $\mathcal{𝒜}$ are (uniquely) isomorphic.

The canonical universal enveloping group of the U-amalgam $\mathcal{𝒜}$ is the pair $\left(G\left(\mathcal{𝒜}\right),\stackrel{^}{\tau }\right)$, where $G\left(\mathcal{𝒜}\right)$ is the group given by the presentation

$\begin{array}{cc}\hfill G\left(\mathcal{𝒜}\right):=〈\bigcup _{i\ne j\in I}{G}_{ij}|& \text{all relations in}{G}_{ij},\text{and}{\varphi }_{ij}^{j}\left(x\right)={\varphi }_{kj}^{j}\left(x\right)\hfill \\ & \text{for all}i\ne j\ne k\in I\text{and all}x\in U〉\hfill \end{array}$

and where $\stackrel{^}{\tau }=\left\{{\stackrel{^}{\tau }}_{ij}\mid i\ne j\in I\right\}$ with the canonical homomorphism ${\stackrel{^}{\tau }}_{ij}:{G}_{ij}\to G\left(\mathcal{𝒜}\right)$ for all $i\ne j\in I$. The canonical universal enveloping group of a U-amalgam is a universal enveloping group (cf. [25, Lemma 1.3.2]).

Let U and V be groups and I an index set. Suppose

• $\stackrel{~}{\mathcal{𝒜}}=\left\{{\stackrel{~}{G}}_{ij},{\stackrel{~}{\varphi }}_{ij}^{i}\mid i\ne j\in I\right\}$ is a U -amalgam over I,

• $\mathcal{𝒜}=\left\{{G}_{ij},{\varphi }_{ij}^{i}\mid i\ne j\in I\right\}$ is a V -amalgam over I,

• $\alpha =\left\{\pi ,{\rho }^{i},{\alpha }_{ij}\mid i\ne j\in I\right\}$ is an amalgam epimorphism $\stackrel{~}{\mathcal{𝒜}}\to \mathcal{𝒜}$,

• $\left(G,\tau \right)$ with $\tau =\left\{{\tau }_{ij}\mid i\ne j\in I\right\}\right)$ is an enveloping group of $\mathcal{𝒜}$.

Then the following hold:

• (a)

There is a unique enveloping group $\left(G,\stackrel{~}{\tau }\right)$, $\stackrel{~}{\tau }=\left\{{\stackrel{~}{\tau }}_{ij}\mid i\ne j\in I\right\}$ , of $\stackrel{~}{\mathcal{𝒜}}$ such that the following diagram commutes for all $i\ne j\in I$ :

• (b)

Suppose $\left(\stackrel{~}{G},\stackrel{~}{\tau }\right)$, $\stackrel{~}{\tau }=\left\{{\stackrel{~}{\tau }}_{ij}\mid i\ne j\in I\right\}$ , is a universal enveloping group of $\stackrel{~}{\mathcal{𝒜}}$ . Then there is a unique epimorphism $\stackrel{^}{\alpha }:\stackrel{~}{G}\to G$ such that the following diagram commutes for all $i\ne j\in I$ :

• (c)

If α is an isomorphism and $\left(G,\tau \right)$ is a universal enveloping group, then $\stackrel{^}{\alpha }$ is also an isomorphism.

Proof.

(a) Let $i\ne j\in I$. Since ${\alpha }_{ij}$ is an epimorphism, we must have

${\stackrel{~}{\tau }}_{ij}:={\tau }_{\pi \left(i\right)\pi \left(j\right)}\circ {\alpha }_{ij}$

for the diagrams to commute; the claimed uniqueness follows. The fact that ${\alpha }_{ij}$ is an epimorphism also implies

${\stackrel{~}{\tau }}_{ij}\left({\stackrel{~}{G}}_{ij}\right)=\left({\tau }_{\pi \left(i\right)\pi \left(j\right)}\circ {\alpha }_{ij}\right)\left({\stackrel{~}{G}}_{ij}\right)={\tau }_{\pi \left(i\right)\pi \left(j\right)}\left({G}_{\pi \left(i\right)\pi \left(j\right)}\right),$

and so

$G=〈{\tau }_{ij}\left({\stackrel{~}{G}}_{ij}\right)〉=〈{\stackrel{~}{\tau }}_{ij}\left({\stackrel{~}{G}}_{ij}\right)〉.$

Moreover, for $i\ne j\ne k\in I$ we find

${\stackrel{~}{\tau }}_{ij}\circ {\stackrel{~}{\varphi }}_{ij}^{j}={\tau }_{\pi \left(i\right)\pi \left(j\right)}\circ {\alpha }_{ij}\circ {\stackrel{~}{\varphi }}_{ij}^{j}$$={\tau }_{\pi \left(i\right)\pi \left(j\right)}\circ {\varphi }_{\pi \left(i\right)\pi \left(j\right)}^{\pi \left(j\right)}\circ {\rho }^{\pi \left(j\right)}$$={\tau }_{\pi \left(k\right)\pi \left(j\right)}\circ {\varphi }_{\pi \left(k\right)\pi \left(j\right)}^{\pi \left(j\right)}\circ {\rho }^{\pi \left(j\right)}$$={\tau }_{\pi \left(k\right)\pi \left(j\right)}\circ {\alpha }_{kj}\circ {\stackrel{~}{\varphi }}_{kj}^{j}$$={\stackrel{~}{\tau }}_{kj}\circ {\stackrel{~}{\varphi }}_{kj}^{j}.$

Hence $\left(G,\left\{{\stackrel{~}{\tau }}_{ij}\right\}\right)$ is indeed an enveloping group of $\stackrel{~}{\mathcal{𝒜}}$.

(b) On the one hand, by (a) the lower left triangle in the following diagram commutes:

On the other hand, by the definition of universal enveloping group there is a unique epimorphism $\stackrel{^}{\alpha }$ making the upper right triangle commute. The claim follows.

(c) This follows from part (b) by interchanging the roles of ${G}_{\pi \left(i\right)\pi \left(j\right)}$, G and ${\stackrel{~}{G}}_{ij}$, $\stackrel{~}{G}$. ∎

We denote the situation in Lemma 7(a) by the commutative diagram

and the situation in Lemma 7(b) by the commutative diagram

The following proposition will be crucial throughout this article. The typical situation in our applications will be $U=\mathrm{SO}\left(2\right)$, $\stackrel{~}{U}=\mathrm{Spin}\left(2\right)$, $\stackrel{~}{V}=\left\{±1\right\}$.

Let U, $\stackrel{\mathrm{~}}{U}$ and $\stackrel{\mathrm{~}}{V}\mathrm{\le }\stackrel{\mathrm{~}}{U}$ be groups and I an index set. Suppose

• $\stackrel{~}{\mathcal{𝒜}}=\left\{{\stackrel{~}{G}}_{ij},{\stackrel{~}{\varphi }}_{ij}^{i}\mid i\ne j\in I\right\}$ is a $\stackrel{~}{U}$ -amalgam over I such that

${\stackrel{~}{G}}_{ij}=〈{\stackrel{~}{\varphi }}_{ij}^{i}\left(\stackrel{~}{U}\right),{\stackrel{~}{\varphi }}_{ij}^{j}\left(\stackrel{~}{U}\right)〉,$

• $\mathcal{𝒜}=\left\{{G}_{ij},{\varphi }_{ij}^{i}\mid i\ne j\in I\right\}$ is a U -amalgam over I,

• $\alpha =\left\{\pi ,{\rho }^{i},{\alpha }_{ij}\mid i\ne j\in I\right\}$ is an amalgam epimorphism $\stackrel{~}{\mathcal{𝒜}}\to \mathcal{𝒜}$,

• $\left(\stackrel{~}{G},\stackrel{~}{\tau }\right)$ with $\stackrel{~}{\tau }=\left\{{\stackrel{~}{\tau }}_{ij}\mid i\ne j\in I\right\}\right)$ is a universal enveloping group of $\stackrel{~}{\mathcal{𝒜}}$,

• $\left(G,\tau \right)$ with $\tau =\left\{{\tau }_{ij}\mid i\ne j\in I\right\}\right)$ is a universal enveloping group of $\mathcal{𝒜}$,

• $\stackrel{^}{\alpha }:\stackrel{~}{G}\to G$ is the epimorphism induced by α via the commutative diagrams ( $i\ne j\in I$ )

as in Lemma 7 (b).

For $i\mathrm{,}j\mathrm{\in }I$, with $i\mathrm{\ne }j$ define ${Z}_{i\mathit{}j}^{i}\mathrm{:=}{\stackrel{\mathrm{~}}{\varphi }}_{i\mathit{}j}^{i}\mathit{}\mathrm{\left(}\stackrel{\mathrm{~}}{V}\mathrm{\right)}$ and ${Z}_{i\mathit{}j}^{\stackrel{\mathrm{~}}{\varphi }}\mathrm{:=}\mathrm{〈}{Z}_{i\mathit{}j}^{i}\mathrm{,}{Z}_{i\mathit{}j}^{j}\mathrm{〉}$, as well as ${A}_{i\mathit{}j}\mathrm{:=}\mathrm{ker}\mathit{}\mathrm{\left(}{\alpha }_{i\mathit{}j}\mathrm{\right)}$. Then if

${A}_{ij}\le {Z}_{ij}^{\stackrel{~}{\varphi }}\le Z\left({\stackrel{~}{G}}_{ij}\right),$

it follows that $\stackrel{\mathrm{~}}{G}$ is a central extension of G by $N\mathrm{:=}\mathrm{〈}{\stackrel{\mathrm{~}}{\tau }}_{i\mathit{}j}\mathit{}\mathrm{\left(}{A}_{i\mathit{}j}\mathrm{\right)}\mathrm{\mid }i\mathrm{\ne }j\mathrm{\in }I\mathrm{〉}$.

In this situation the epimorphism $\alpha \mathrm{:}\stackrel{\mathrm{~}}{\mathcal{A}}\mathrm{\to }\mathcal{A}$ is called an $\mathrm{|}N\mathrm{|}$-fold central extension of amalgams.

Proof.

We proceed by proving the following two assertions:

• (a)

$N\le Z\left(\stackrel{~}{G}\right)$,

• (b)

$\stackrel{~}{G}/N\cong G$.

Consider the following commutative diagram:

For $i\ne j\in I$ set

${Z}_{i}:={\stackrel{~}{\tau }}_{ij}\left({Z}_{ij}^{i}\right)=\left({\stackrel{~}{\tau }}_{ij}\circ {\stackrel{~}{\varphi }}_{ij}^{i}\right)\left(\stackrel{~}{V}\right)\mathit{ }\text{and}\mathit{ }{\stackrel{~}{G}}_{i}:=\left({\stackrel{~}{\tau }}_{ij}\circ {\stackrel{~}{\varphi }}_{ij}^{i}\right)\left(\stackrel{~}{U}\right).$

The hypothesis implies

${\stackrel{~}{\tau }}_{ij}\left({\stackrel{~}{G}}_{ij}\right)=〈{\stackrel{~}{G}}_{i},{\stackrel{~}{G}}_{j}〉\mathit{ }\text{and}\mathit{ }\stackrel{~}{G}=〈{\stackrel{~}{\tau }}_{ij}\left({\stackrel{~}{G}}_{ij}\right)\mid i\ne j\in I〉=〈{\stackrel{~}{G}}_{i}\mid i\in I〉.$

Moreover,

${Z}_{i}={\stackrel{~}{\tau }}_{ij}\left({Z}_{ij}^{i}\right)\le {\stackrel{~}{\tau }}_{ij}\left(Z\left({\stackrel{~}{G}}_{ij}\right)\right)\le Z\left({\stackrel{~}{\tau }}_{ij}\left({\stackrel{~}{G}}_{ij}\right)\right)=Z\left(〈{\stackrel{~}{G}}_{i},{\stackrel{~}{G}}_{j}〉\right),$

whence ${Z}_{i}$ centralises ${\stackrel{~}{G}}_{j}$ for all $i,j\in I$. Since $\stackrel{~}{G}$ is generated by the ${\stackrel{~}{G}}_{i}$, one has

$〈{Z}_{i}\mid i\in I〉\le Z\left(〈{\stackrel{~}{G}}_{i}\mid i\in I〉\right)=Z\left(\stackrel{~}{G}\right).$

Therefore

$N=〈{\stackrel{~}{\tau }}_{ij}\left({A}_{ij}\right)\mid i\ne j\in I〉\le 〈{\stackrel{~}{\tau }}_{ij}\left({Z}_{ij}^{i}\right)\mid i\ne j\in I〉=〈{Z}_{i}\mid i\in I〉\le Z\left(\stackrel{~}{G}\right),$

i.e., (a) holds.

Commutativity of the diagram implies $N\le \mathrm{ker}\left(\stackrel{^}{\alpha }\right)$ and so the homomorphism theorem yields an epimorphism $\stackrel{~}{G}/N\to G$, $gN↦\stackrel{^}{\alpha }\left(g\right)$. We construct an inverse map by exploiting that G and $\stackrel{~}{G}$ are universal enveloping groups of $\mathcal{𝒜}$, resp. $\stackrel{~}{\mathcal{𝒜}}$, in order to show that this epimorphism actually is an isomorphism. Indeed, for $g\in {G}_{ij}$, let $\stackrel{~}{g}\in {\alpha }_{ij}^{-1}\left(g\right)$. Then

${\stackrel{~}{\tau }}_{ij}\left({\alpha }_{ij}^{-1}\left(g\right)\right)={\stackrel{~}{\tau }}_{ij}\left(\stackrel{~}{g}{A}_{ij}\right)\le {\stackrel{~}{\tau }}_{ij}\left(\stackrel{~}{g}\right)N={\stackrel{~}{\tau }}_{ij}\left({\alpha }_{ij}^{-1}\left(g\right)\right)N\in \stackrel{~}{G}/N.$

Thus one obtains a well-defined homomorphism

${\stackrel{^}{\tau }}_{\pi \left(i\right)\pi \left(j\right)}:{G}_{\pi \left(i\right)\pi \left(j\right)}\to \stackrel{~}{G}/N,g↦{\stackrel{~}{\tau }}_{ij}\left({\alpha }_{ij}^{-1}\left(g\right)\right)N.$

Then $\left(\stackrel{~}{G}/N,\left\{{\stackrel{^}{\tau }}_{ij}\right\}\right)$ is an enveloping group for $\mathcal{𝒜}$. In particular, for $u\in U$ and $i\ne j\ne k\in I$ one has

$\left({\stackrel{^}{\tau }}_{\pi \left(i\right)\pi \left(j\right)}\circ {\varphi }_{\pi \left(i\right)\pi \left(j\right)}^{\pi \left(j\right)}\right)\left(u\right)={\stackrel{~}{\tau }}_{ij}\left({\alpha }_{ij}^{-1}\left({\varphi }_{\pi \left(i\right)\pi \left(j\right)}^{\pi \left(j\right)}\left(u\right)\right)\right)$$={\stackrel{~}{\tau }}_{ij}\left({\stackrel{~}{\varphi }}_{ij}^{j}\left({\left({\rho }^{j}\right)}^{-1}\left(u\right)\right)\right)$$={\stackrel{~}{\tau }}_{kj}\left({\stackrel{~}{\varphi }}_{kj}^{j}\left({\left({\rho }^{j}\right)}^{-1}\left(u\right)\right)\right)$$={\stackrel{~}{\tau }}_{kj}\left({\alpha }_{kj}^{-1}\left({\varphi }_{\pi \left(k\right)\pi \left(j\right)}^{\pi \left(j\right)}\left(u\right)\right)\right)$$=\left({\stackrel{^}{\tau }}_{\pi \left(k\right)\pi \left(j\right)}\circ {\varphi }_{\pi \left(k\right)\pi \left(j\right)}^{\pi \left(j\right)}\right)\left(u\right).$

Since $\left(G,\left\{{\tau }_{ij}\right\}\right)$ is a universal enveloping group of $\mathcal{𝒜}$, there exists a unique epimorphism $\beta :G\to \stackrel{~}{G}/N$ such that for $i\ne j\in I$ one has

$\beta \circ {\tau }_{ij}={\stackrel{^}{\tau }}_{ij}.$

By the definition of $\stackrel{^}{\alpha }$ and ${\stackrel{^}{\tau }}_{ij}$ one finds

$\stackrel{^}{\alpha }\circ {\stackrel{^}{\tau }}_{ij}={\tau }_{ij}.$

Therefore

$\left(\beta \circ \stackrel{^}{\alpha }\right)\circ {\stackrel{^}{\tau }}_{ij}={\stackrel{^}{\tau }}_{ij}\mathit{ }\text{and}\mathit{ }\left(\stackrel{^}{\alpha }\circ \beta \right)\circ {\tau }_{ij}={\tau }_{ij}.$

But $\left(G,\tau \right)$ and $\left(\stackrel{~}{G},\stackrel{~}{\tau }\right)$ are universal enveloping groups; their uniqueness property implies that $\beta \circ \stackrel{^}{\alpha }={id}_{\stackrel{~}{G}/N}$ and $\stackrel{^}{\alpha }\circ \beta ={id}_{G}$ and hence as claimed $\stackrel{~}{G}/N\cong G$. We have shown assertion (b). ∎

4 Cartan matrices and Dynkin diagrams

In this section we recall the concepts of Cartan matrices and Dynkin diagrams. For a thorough introduction see [26, Chapter 4] and [34, Section 7.1].

Let I be a non-empty set. A generalised Cartan matrix over I is a matrix $A={\left(a\left(i,j\right)\right)}_{i,j\in I}$ such that for all $i\ne j\in I$,

• (a)

$a\left(i,i\right)=2$,

• (b)

$a\left(i,j\right)$ is a non-positive integer,

• (c)

if $a\left(i,j\right)=0$ then $a\left(j,i\right)=0$.

The matrix A is of two-spherical type if $a\left(i,j\right)a\left(j,i\right)\in \left\{0,1,2,3\right\}$ for all $i,j\in I$ with $i\ne j$.

A Dynkin diagram (or short: diagram) is a graph Π with vertex set $V\left(\mathrm{\Pi }\right)$ and edge set $E\left(\mathrm{\Pi }\right)\subseteq \left(\genfrac{}{}{0pt}{}{V\left(\mathrm{\Pi }\right)}{2}\right)$ such that each edge has an edge valency of 1, 2, 3 or $\mathrm{\infty }$ and, in addition, edges with valency 2 or 3 are directed. If $\left\{v,w\right\}\in E\left(\mathrm{\Pi }\right)$ is directed from v to w, we write $v\to w$. Let ${E}_{0}\left(\mathrm{\Pi }\right):=\left(\genfrac{}{}{0pt}{}{V\left(\mathrm{\Pi }\right)}{2}\right)\setminus E\left(\mathrm{\Pi }\right)$, and let ${E}_{1}\left(\mathrm{\Pi }\right)$, ${E}_{2}\left(\mathrm{\Pi }\right)$, ${E}_{3}\left(\mathrm{\Pi }\right)$, resp. ${E}_{\mathrm{\infty }}\left(\mathrm{\Pi }\right)$ be the subsets of $E\left(\mathrm{\Pi }\right)$ of edges of valency 1, 2, 3, resp. $\mathrm{\infty }$. The elements of ${E}_{1}\left(\mathrm{\Pi }\right)$, ${E}_{2}\left(\mathrm{\Pi }\right)$, ${E}_{3}\left(\mathrm{\Pi }\right)$ are called edges of type ${A}_{2}$, ${\mathrm{C}}_{2}$ resp. ${\mathrm{G}}_{2}$. The diagram Π is irreducible if it is connected as a graph, it is simply laced if all edges have valency 1, it is doubly laced if all edges have valency 1 or 2, and it is two-spherical if no edge has valency $\mathrm{\infty }$. If $V\left(\mathrm{\Pi }\right)$ is finite, then a labelling of Π is a bijection $\sigma :I\to V\left(\mathrm{\Pi }\right)$, where $I:=\left\{1,\mathrm{\dots },|V\left(\mathrm{\Pi }\right)|\right\}$.

Throughout this text, we assume all diagrams to have finite vertex set.

Let I be a non-empty set and $A={\left(a\left(i,j\right)\right)}_{i,j\in I}$ a two-spherical generalised Cartan matrix. Then this induces a two-spherical Dynkin diagram $\mathrm{\Pi }\left(A\right)$ with vertex set $V:=I$ as follows: For $i,j\in I$ with $i\ne j$, there is an edge between i and j if and only if $a\left(i,j\right)\ne 0$. The valency of the edge then is ${v}_{ij}:=a\left(i,j\right)a\left(j,i\right)\in \left\{1,2,3\right\}$. If ${v}_{ij}>1$, then the edge is directed $i←j$ if and only if $a\left(i,j\right)=-{v}_{ij}<-1=a\left(j,i\right)$.

Conversely, given a two-spherical Dynkin diagram Π with vertex set V, we obtain a two-spherical generalised Cartan matrix $A\left(\mathrm{\Pi }\right):={\left(a\left(i,j\right)\right)}_{i,j\in I}$ over $I:=V$ by setting for $i\ne j\in I$,

$a\left(i,i\right):=2,a\left(i,j\right):=\left\{\begin{array}{cc}0,\hfill & \text{if}\left\{i,j\right\}\notin E\left(\mathrm{\Pi }\right),\hfill \\ -2,\hfill & \text{if}\left\{i,j\right\}\in {E}_{2}\left(\mathrm{\Pi }\right)\text{and}i←j,\hfill \\ -3,\hfill & \text{if}\left\{i,j\right\}\in {E}_{3}\left(\mathrm{\Pi }\right)\text{and}i←j,\hfill \\ -1,\hfill & \text{otherwise}.\hfill \end{array}$

These two operations are inverse to each other, i.e., we have $\mathrm{\Pi }\left(A\left(\mathrm{\Pi }\right)\right)=\mathrm{\Pi }$ and $A\left(\mathrm{\Pi }\left(A\right)\right)=A$.

Note that in [4, Definition VI.§1.3, p. 167] the opposite convention for the entries of the Cartan matrix is used.

If the generalised Cartan matrix A is not of two-spherical type, it is nevertheless possible to associate a Dynkin diagram $\mathrm{\Pi }\left(A\right)$ to it by labelling the edge between i and j with $\mathrm{\infty }$ whenever $a\left(i,j\right)a\left(j,i\right)\ge 4$. In this case it is, of course, not possible to reconstruct the values of $a\left(i,j\right)$ and $a\left(j,i\right)$ from the diagram Π.

Therefore, by convention, in this article for each edge between i and j with label $\mathrm{\infty }$ we consider the values of $a\left(i,j\right)$ and $a\left(j,i\right)$ as part of the augmented Dynkin diagram: write $-a\left(i,j\right)$ between the vertex i and the $\mathrm{\infty }$ label and $-a\left(j,i\right)$ between the vertex j and the $\mathrm{\infty }$ label. In addition, an edge with $\mathrm{\infty }$ label such that $a\left(i,j\right)$ and $a\left(j,i\right)$ have different parity shall be directed $i←j$, if $a\left(i,j\right)$ is even, and $i\to j$, if $a\left(i,j\right)$ is odd. See Figure 1 for an example.

Figure 1

An augmented Dynkin diagram.

5 The groups $\mathrm{SO}\left(n\right)$ and $\mathrm{O}\left(n\right)$

In this section we fix notation concerning the compact real orthogonal groups.

Given a quadratic space $\left(𝕂,V,q\right)$ with ${dim}_{𝕂}V<\mathrm{\infty }$, we set

$\mathrm{O}\left(q\right):=\left\{a\in \mathrm{GL}\left(V\right)\mid q\left(av\right)=q\left(v\right)\text{for all}v\in V\right\},$$\mathrm{SO}\left(q\right):=\mathrm{O}\left(q\right)\cap \mathrm{SL}\left(V\right).$

Given $n\in ℕ$, let ${q}_{n}:{ℝ}^{n}\to ℝ$, $x↦{\sum }_{i=1}^{n}{x}_{i}^{2}$ be the standard quadratic form on ${ℝ}^{n}$, and

$\mathrm{O}\left(n\right):=\left\{a\in {\mathrm{GL}}_{n}\left(ℝ\right)\mid a{a}^{t}={E}_{n}\right\}\cong \mathrm{O}\left({q}_{n}\right)=\mathrm{O}\left(-{q}_{n}\right),$$\mathrm{SO}\left(n\right):=\mathrm{O}\left(n\right)\cap {\mathrm{SL}}_{n}\left(ℝ\right)\cong \mathrm{SO}\left({q}_{n}\right)=\mathrm{SO}\left(-{q}_{n}\right)⊴\mathrm{O}\left({q}_{n}\right)=\mathrm{O}\left(-{q}_{n}\right).$

Since an element of $\mathrm{O}\left(n\right)$ has determinant 1 or -1, we have $\left[\mathrm{O}\left(n\right):\mathrm{SO}\left(n\right)\right]=2$.

Let $n\in ℕ$ and let $\mathcal{ℰ}=\left({e}_{1},\mathrm{\dots },{e}_{n}\right)$ be the standard basis of ${ℝ}^{n}$. Given a subset $I\subseteq \left\{1,\mathrm{\dots },n\right\}$, we set

${\mathcal{ℰ}}_{I}:=\left\{{e}_{i}\mid i\in I\right\},{V}_{I}:={〈{\mathcal{ℰ}}_{I}〉}_{ℝ}\le {ℝ}^{n},{q}_{I}:=q_{n}{}_{|{V}_{I}}:{V}_{I}\to ℝ.$

There are canonical isomorphisms

${M}_{\mathcal{ℰ}}:End\left({ℝ}^{n}\right)\to {\mathrm{M}}_{n}\left(ℝ\right),a↦{M}_{\mathcal{ℰ}}\left(a\right)$

and

${M}_{{\mathcal{ℰ}}_{I}}:End\left({V}_{I}\right)\to {M}_{|I|}\left(ℝ\right),a↦{M}_{{\mathcal{ℰ}}_{I}}\left(a\right)$

that map an endomorphism into its transformation matrix with respect to the standard basis $\mathcal{ℰ}$, resp. the basis ${\mathcal{ℰ}}_{I}$. Moreover, there is a canonical embedding

${\epsilon }_{I}:\mathrm{O}\left({q}_{I}\right)\to \mathrm{O}\left({q}_{n}\right),$

inducing a canonical embedding

${M}_{\mathcal{ℰ}}\circ {\epsilon }_{I}\circ {M}_{{\mathcal{ℰ}}_{I}}^{-1}:\mathrm{O}\left(|I|\right)\to \mathrm{O}\left(n\right),$

which, by slight abuse of notation, we also denote by ${\epsilon }_{I}$. We will furthermore use the same symbol for the (co)restriction of ${\epsilon }_{I}$ to $\mathrm{SO}\left(\cdot \right)$. The most important application of this map in this article is for $|I|=2$ with $I=\left\{i,j\right\}$ providing the map

${\epsilon }_{ij}:\mathrm{SO}\left(2\right)\to \mathrm{SO}\left(n\right).$

6 The groups $\mathrm{Spin}\left(n\right)$ and $\mathrm{Pin}\left(n\right)$

In this section we recall the compact real spin and pin groups. For a thorough treatment we refer to [30, 13, 32].

Let $\left(ℝ,V,q\right)$ be a quadratic space and let $T\left(V\right)={\oplus }_{n\ge 0}{V}^{\otimes n}$ be the tensor algebra of V. The identity ${V}^{\otimes 0}=ℝ$ provides a ring monomorphism $ℝ\to T\left(V\right)$, and the identity ${V}^{\otimes 1}=V$ a vector space monomorphism $V\to T\left(V\right)$; these allow one to identify $ℝ$, V with their respective images in $T\left(V\right)$. For

$\Im \left(q\right):=〈v\otimes v-q\left(v\right)\mid v\in V〉$

define the Clifford algebra of $q$ as

$\mathrm{Cl}\left(q\right):=T\left(V\right)/\Im \left(q\right).$

Moreover, let

$\mathrm{Cl}{\left(q\right)}^{*}:=\left\{x\in \mathrm{Cl}\left(q\right)\mid \text{there exists}y\in \mathrm{Cl}\left(q\right)\text{such that}xy=1\right\}.$

The transposition map is the involution

$\tau :\mathrm{Cl}\left(q\right)\to \mathrm{Cl}\left(q\right)\mathit{ }\text{induced by}\mathit{ }{v}_{1}\mathrm{\cdots }{v}_{k}↦{v}_{k}\mathrm{\cdots }{v}_{1},{v}_{i}\in V,$

cf. [32, Section 2.2.6], [13, Proposition 1.1]. The parity automorphism is the map

$\mathrm{\Pi }:\mathrm{Cl}\left(q\right)\to \mathrm{Cl}\left(q\right)\mathit{ }\text{given by}\mathit{ }{v}_{1}\mathrm{\cdots }{v}_{k}↦{\left(-1\right)}^{k}\cdot {v}_{1}\mathrm{\cdots }{v}_{k},{v}_{i}\in V,$

cf. [32, Section 2.2.2, Section 3.1.1], [13, Proposition 1.2]. We set

${\mathrm{Cl}}^{0}\left(q\right):=\left\{x\in \mathrm{Cl}\left(q\right)\mid \mathrm{\Pi }\left(x\right)=x\right\}$

and

${\mathrm{Cl}}^{1}\left(q\right):=\left\{x\in \mathrm{Cl}\left(q\right)\mid \mathrm{\Pi }\left(x\right)=-x\right\},$

which yields a ${ℤ}_{2}$-grading of $\mathrm{Cl}\left(q\right)$, i.e.,

$\mathrm{Cl}\left(q\right)={\mathrm{Cl}}^{0}\left(q\right)\oplus {\mathrm{Cl}}^{1}\left(q\right)$

and

${\mathrm{Cl}}^{i}\left(q\right){\mathrm{Cl}}^{j}\left(q\right)\subseteq {\mathrm{Cl}}^{i+j}\left(q\right)\mathit{ }\text{for}i,j\in {ℤ}_{2}.$

Furthermore, following [13, Section 3.1], we define the Clifford conjugation

$\sigma :\mathrm{Cl}\left(q\right)\to \mathrm{Cl}\left(q\right),x↦\overline{x}:=\tau \mathrm{\Pi }\left(x\right)=\mathrm{\Pi }\tau \left(x\right),$

and the spinor norm

$N:\mathrm{Cl}\left(q\right)\to \mathrm{Cl}\left(q\right),x↦x\overline{x}.$

In the following, $\left(ℝ,V,q\right)$ is an anisotropic quadratic space such that ${dim}_{ℝ}V<\mathrm{\infty }$.

Given $x\in \mathrm{Cl}{\left(q\right)}^{*}$, the map

${\rho }_{x}:\mathrm{Cl}\left(q\right)\to \mathrm{Cl}\left(q\right),y↦\mathrm{\Pi }\left(x\right)y{x}^{-1}$

is the twisted conjugation with respect to $x$ . Using the canonical identification of V with its image in $\mathrm{Cl}\left(q\right)$, we define

$\mathrm{\Gamma }\left(q\right):=\left\{x\in \mathrm{Cl}{\left(q\right)}^{*}\mid {\rho }_{x}\left(v\right)\in V\text{for all}v\in V\right\}$

to be the Clifford group with respect to $q$ , cf. [32, Section 3.1.1], [13, Definition 1.4]. We obtain a representation

$\rho :\mathrm{\Gamma }\left(q\right)\to \mathrm{GL}\left(V\right),x↦{\rho }_{x},$

which is the twisted adjoint representation.

Given $n\in ℕ$ and $V={ℝ}^{n}$, we set

$\mathrm{Cl}\left(n\right):=\mathrm{Cl}\left(-{q}_{n}\right)\mathit{ }\text{and}\mathit{ }\mathrm{\Gamma }\left(n\right):=\mathrm{\Gamma }\left(-{q}_{n}\right).$

Recall that ${q}_{n}$ is defined to be the standard quadratic form on ${ℝ}^{n}$ (cf. Definition 1) and note that in the literature one can also find the opposite sign convention.

(a) Let $n\in ℕ$ and let ${e}_{1},\mathrm{\dots },{e}_{n}$ be the standard basis of ${ℝ}^{n}$. Then the following hold in $\mathrm{Cl}\left(n\right)$ for $1\le i\ne j\le n$,

${e}_{i}^{2}=-1,$${e}_{i}{e}_{j}=-{e}_{j}{e}_{i},$${\left({e}_{i}{e}_{j}\right)}^{2}=-1.$

The first identity is immediate from the definition. The second identity follows from polarization, as in the tensor algebra $T\left({ℝ}^{n}\right)$ one has

$\Im \left({q}_{n}\right)\ni \left({e}_{i}+{e}_{j}\right)\otimes \left({e}_{i}+{e}_{j}\right)-q\left({e}_{i}+{e}_{j}\right)$$={e}_{i}\otimes {e}_{i}+{e}_{i}\otimes {e}_{j}+{e}_{j}\otimes {e}_{i}+{e}_{j}\otimes {e}_{j}-q\left({e}_{i}\right)-q\left({e}_{j}\right)-2b\left({e}_{i},{e}_{j}\right)$$={e}_{i}\otimes {e}_{j}+{e}_{j}\otimes {e}_{i},$

where $b\left(\cdot ,\cdot \right)$ denotes the bilinear form associated to ${q}_{n}$. The third identity is immediate from the first two.

(b) One has ${\mathrm{Cl}}^{0}\left(3\right)\cong ℍ$, where $ℍ$ denotes the quaternions. Indeed, given a basis ${e}_{1}$, ${e}_{2}$, ${e}_{3}$ of ${ℝ}^{3}$, a basis of ${\mathrm{Cl}}^{0}\left(3\right)$, considered as an $ℝ$-vector space, is given by 1, ${e}_{1}{e}_{2}$, ${e}_{2}{e}_{3}$, ${e}_{3}{e}_{1}$. By (a) the latter three basis elements square to -1 and anticommute with one another. Note, furthermore, that under this isomorphism the Clifford conjugation is transformed into the standard involution of the quaternions, i.e., the conjugation obtained from the Cayley–Dickson construction. Consequently, the spinor norm is transformed into the norm of the quaternions.

The map $N\mathrm{:}\mathrm{Cl}\mathit{}\mathrm{\left(}q\mathrm{\right)}\mathrm{\to }\mathrm{Cl}\mathit{}\mathrm{\left(}q\mathrm{\right)}$ induces a homomorphism

$N:\mathrm{\Gamma }\left(q\right)\to {ℝ}^{*}$

such that

$N\left(\mathrm{\Pi }\left(x\right)\right)=N\left(x\right)\mathit{ }\mathit{\text{for all}}x\in \mathrm{\Gamma }\left(q\right).$

Proof.

Cf. [13, Proposition 1.9]. ∎

The group

$\mathrm{Pin}\left(q\right):=\left\{x\in \mathrm{\Gamma }\left(q\right)\mid N\left(x\right)=1\right\}\le \mathrm{\Gamma }\left(q\right)$

is the pin group with respect to $q$ , and

$\mathrm{Spin}\left(q\right):=\mathrm{Pin}\left(q\right)\cap {\mathrm{Cl}}^{0}\left(q\right)\le \mathrm{Pin}\left(q\right)$

is the spin group with respect to $q$ . By Lemma 6 and the ${ℤ}_{2}$-grading of $\mathrm{Cl}\left(q\right)$, the sets $\mathrm{Pin}\left(q\right)$ and $\mathrm{Spin}\left(q\right)$ are indeed subgroups of $\mathrm{\Gamma }\left(q\right)$. Given $n\in ℕ$, define

$\mathrm{Pin}\left(n\right):=\mathrm{Pin}\left(-{q}_{n}\right)\mathit{ }\text{and}\mathit{ }\mathrm{Spin}\left(n\right):=\mathrm{Spin}\left(-{q}_{n}\right).$

The following hold:

• (a)

One has $\left[\mathrm{Pin}\left(q\right):\mathrm{Spin}\left(q\right)\right]=2$ and $\mathrm{Spin}\left(q\right)={\rho }^{-1}\left(\mathrm{SO}\left(q\right)\right)$.

• (b)

The twisted adjoint representation $\rho :\mathrm{\Gamma }\left(q\right)\to \mathrm{GL}\left(V\right)$ induces an epimorphism $\rho :\mathrm{Pin}\left(q\right)\to \mathrm{O}\left(q\right)$ . In particular, given $n\in ℕ$ , we obtain epimorphisms

${\rho }_{n}:={M}_{\mathcal{ℰ}}\circ \rho :\mathrm{Pin}\left(n\right)\to \mathrm{O}\left(n\right),{\rho }_{n}:={M}_{\mathcal{ℰ}}\circ \rho :\mathrm{Spin}\left(n\right)\to \mathrm{SO}\left(n\right)$

with $\mathrm{ker}\left({\rho }_{n}\right)=\left\{±1\right\}$ in both cases.

• (c)

The group $\mathrm{Spin}\left(q\right)$ is a double cover of the group $\mathrm{SO}\left(q\right)$.

Proof.

See [13, Theorem 1.11]. ∎

(a) By slight abuse of notation, suppressing the choice of basis, we will also sometimes denote the map ${\rho }_{n}$ by ρ.

(b) Let ${H}_{1}\le \mathrm{Spin}\left(n\right)$ and ${H}_{2}\le \mathrm{Pin}\left(n\right)$ be such that $-1\in {H}_{1}$ and $-1\in {H}_{2}$, respectively, and let ${\stackrel{~}{H}}_{i}:={\rho }_{n}\left({H}_{i}\right)$. Then we have ${H}_{i}={\rho }_{n}^{-1}\left({\stackrel{~}{H}}_{i}\right)$. We will explicitly determine these groups for some canonical subgroups of $\mathrm{SO}\left(n\right)$ and $\mathrm{O}\left(n\right)$.

Let $n\mathrm{\in }\mathrm{N}$, let $I\mathrm{\subseteq }\mathrm{\left\{}\mathrm{1}\mathrm{,}\mathrm{\dots }\mathrm{,}n\mathrm{\right\}}$, and let ${\stackrel{\mathrm{~}}{\epsilon }}_{I}\mathrm{:}\mathrm{Cl}\mathit{}\mathrm{\left(}\mathrm{-}{q}_{I}\mathrm{\right)}\mathrm{\to }\mathrm{Cl}\mathit{}\mathrm{\left(}\mathrm{-}{q}_{n}\mathrm{\right)}$ be the natural embedding of algebras afforded by the inclusion

$\left\{{e}_{i}\mid i\in I\right\}\to \left\{{e}_{i}\mid 1\le i\le n\right\}$

of bases of ${V}_{I}$, resp. V. Then ${\stackrel{\mathrm{~}}{\epsilon }}_{I}$ restricts and corestricts to an embedding

$\mathrm{Pin}\left(-{q}_{I}\right)\to \mathrm{Pin}\left(-{q}_{n}\right)$

of groups such that the following diagram commutes:

In analogy to Notation 2 we will use the same symbol for the (co)restriction of ${\stackrel{~}{\epsilon }}_{I}$ to $\mathrm{Spin}\left(\cdot \right)$. The most important application of this map in this article is for $|I|=2$ with $I=\left\{i,j\right\}$ providing the map

${\stackrel{~}{\epsilon }}_{ij}:\mathrm{Spin}\left(2\right)\to \mathrm{Spin}\left(n\right).$

Proof.

Let $x\in \mathrm{\Gamma }\left(-{q}_{I}\right)$. By definition,

${\rho }_{{\stackrel{~}{\epsilon }}_{I}\left(x\right)}\left(v\right)=\left({\epsilon }_{I}\circ \rho \right)\left(v\right)\in {V}_{I}\subseteq V={ℝ}^{n}\mathit{ }\text{for all}v\in {V}_{I}.$

Since ${e}_{i}{e}_{j}=-{e}_{j}{e}_{i}$ for all $i\ne j\in I$ by Remark 5 (a), for each $ℝ$-basis vector $y={e}_{{j}_{1}}\mathrm{\cdots }{e}_{{j}_{k}}$ of ${\stackrel{~}{\epsilon }}_{I}\left(\mathrm{Cl}\left(-{q}_{I}\right)\right)$ and all $i\in \left\{1,\mathrm{\dots },n\right\}\I$ one has

$\mathrm{\Pi }\left(y\right){e}_{i}={e}_{i}y.$

Hence

$\mathrm{\Pi }\left({\stackrel{~}{\epsilon }}_{I}\left(x\right)\right){e}_{i}={e}_{i}{\stackrel{~}{\epsilon }}_{I}\left(x\right)$

and thus for all $i\in \left\{1,\mathrm{\dots },n\right\}\I$,

${\rho }_{{\stackrel{~}{\epsilon }}_{I}\left(x\right)}\left({e}_{i}\right)=\mathrm{\Pi }\left({\stackrel{~}{\epsilon }}_{I}\left(x\right)\right){e}_{i}{\stackrel{~}{\epsilon }}_{I}{\left(x\right)}^{-1}={e}_{i}\in {ℝ}^{n}.$

As ${\stackrel{~}{\epsilon }}_{I}\left(\mathrm{Cl}\left(-{q}_{I}\right)\right)$ is generated as an $ℝ$-algebra by the set $\left\{{e}_{i}\mid i\in I\right\}$, we in particular have

$\rho \circ {\stackrel{~}{\epsilon }}_{I}={\epsilon }_{I}\circ \rho .$

Therefore ${\epsilon }_{I}\left(x\right)\in \mathrm{\Gamma }\left(-{q}_{n}\right)$. Finally,

$N\left({\stackrel{~}{\epsilon }}_{I}\left(x\right)\right)={\stackrel{~}{\epsilon }}_{I}\left(N\left(x\right)\right)={\stackrel{~}{\epsilon }}_{I}\left(1\right)=1,$

whence

${\stackrel{~}{\epsilon }}_{I}\left(x\right)\in \mathrm{Pin}\left(-{q}_{n}\right).\mathit{∎}$

Since ${\stackrel{~}{\epsilon }}_{I}\left(\mathrm{Spin}\left(-{q}_{I}\right)\right)=〈{e}_{i}{e}_{j}\mid i\ne j\in I〉\subseteq {\mathrm{Cl}}^{0}\left(-{q}_{n}\right)$, one has

${\stackrel{~}{\epsilon }}_{I}\left(\mathrm{Spin}\left(-{q}_{I}\right)\right)\subseteq \mathrm{Pin}\left(-{q}_{n}\right)\cap {\mathrm{Cl}}^{0}\left(-{q}_{n}\right)=\mathrm{Spin}\left(-{q}_{n}\right).$

Let $n\mathrm{\in }\mathrm{N}$ and $I\mathrm{\subseteq }\mathrm{\left\{}\mathrm{1}\mathrm{,}\mathrm{\dots }\mathrm{,}n\mathrm{\right\}}$. Then

${\rho }_{n}^{-1}\left({\epsilon }_{I}\left(\mathrm{O}\left(|I|\right)\right)\right)={\stackrel{~}{\epsilon }}_{I}\left(\mathrm{Pin}\left(-{q}_{I}\right)\right)\mathit{ }\text{𝑎𝑛𝑑}\mathit{ }{\rho }_{n}^{-1}\left({\epsilon }_{I}\left(\mathrm{SO}\left(|I|\right)\right)\right)={\stackrel{~}{\epsilon }}_{I}\left(\mathrm{Spin}\left(-{q}_{I}\right)\right).$

Proof.

By Lemma 10, one has

${\rho }_{n}{\stackrel{~}{\epsilon }}_{I}\left(\mathrm{Pin}\left(-{q}_{I}\right)\right)={\epsilon }_{I}\left(\mathrm{O}\left(|I|\right)\right)\mathit{ }\text{and}\mathit{ }{\rho }_{n}{\stackrel{~}{\epsilon }}_{I}\left(\mathrm{Spin}\left(-{q}_{I}\right)\right)={\epsilon }_{I}\left(\mathrm{SO}\left(|I|\right)\right),$

thus the assertion results from Remark 9 (b). ∎

Let $n\in ℕ$, let $I\subseteq \left\{1,\mathrm{\dots },n\right\}$ and let $m:=|I|$. Then there exists an isomorphism $i:\mathrm{Pin}\left(m\right)\to \mathrm{Pin}\left(-{q}_{I}\right)$ such that the following diagram commutes:

As in Notation 2 we slightly abuse notation and also write ${\stackrel{~}{\epsilon }}_{I}$ for the map

$id\circ {\stackrel{~}{\epsilon }}_{I}\circ i:\mathrm{Pin}\left(m\right)\to \mathrm{Pin}\left(n\right)$

and ${\epsilon }_{I}$ for the map

${M}_{\epsilon }\circ {\epsilon }_{I}\circ M_{{\epsilon }_{I}}{}^{-1}:\mathrm{O}\left(m\right)\to \mathrm{O}\left(n\right).$

Consequently, we obtain the following commutative diagram:

According to [13, Corollary 1.12], the group $\mathrm{Pin}\left(n\right)$ is generated by the set $\left\{v\in {ℝ}^{n}\mid N\left(v\right)=1\right\}$ and each element of the group $\mathrm{Spin}\left(n\right)$ can be written as a product of an even number of elements from this set. That is, each element $g\in \mathrm{Spin}\left(2\right)$ is of the form

$g=\prod _{i=1}^{2k}\left({a}_{i}{e}_{1}+{b}_{i}{e}_{2}\right)$$=\prod _{i=1}^{k}\left(\left({a}_{2i-1}{a}_{2i}+{b}_{2i-1}{b}_{2i}\right)+\left({a}_{2i-1}{b}_{2i}-{a}_{2i}{b}_{2i-1}\right){e}_{1}{e}_{2}\right)$$=:a+b{e}_{1}{e}_{2}.$

The requirement ${a}_{i}{e}_{1}+{b}_{i}{e}_{2}\in \left\{v\in {ℝ}^{n}\mid N\left(v\right)=1\right\}$ is equivalent to

${a}_{i}^{2}+{b}_{i}^{2}=\left({a}_{i}{e}_{1}+{b}_{i}{e}_{2}\right)\left(-{a}_{i}{e}_{1}-{b}_{i}{e}_{2}\right)$$=\left({a}_{i}{e}_{1}+{b}_{i}{e}_{2}\right)\overline{\left({a}_{i}{e}_{1}+{b}_{i}{e}_{2}\right)}$$=N\left({a}_{i}{e}_{1}+{b}_{i}{e}_{2}\right)=1.$

Moreover,

$1=N\left(g\right)$$=N\left(a+b{e}_{1}{e}_{2}\right)$$=\left(a+b{e}_{1}{e}_{2}\right)\overline{\left(a+b{e}_{1}{e}_{2}\right)}$$=\left(a+b{e}_{1}{e}_{2}\right)\left(a+b{e}_{2}{e}_{1}\right)={a}^{2}+{b}^{2}.$

Certainly, $\mathrm{Spin}\left(2\right)$ contains all elements of the form $a+b{e}_{1}{e}_{2}$ with ${a}^{2}+{b}^{2}=1$, i.e., one obtains

$\mathrm{Spin}\left(2\right)=\left\{\mathrm{cos}\left(\alpha \right)+\mathrm{sin}\left(\alpha \right){e}_{1}{e}_{2}\mid \alpha \in ℝ\right\}.$

One has

${\left(\mathrm{cos}\left(\alpha \right)+\mathrm{sin}\left(\alpha \right){e}_{1}{e}_{2}\right)}^{-1}=\mathrm{cos}\left(\alpha \right)-\mathrm{sin}\left(\alpha \right){e}_{1}{e}_{2}=\mathrm{cos}\left(-\alpha \right)+\mathrm{sin}\left(-\alpha \right){e}_{1}{e}_{2},$

i.e., the map

$ℝ\to \mathrm{Spin}\left(2\right),\alpha ↦\mathrm{cos}\left(\alpha \right)+\mathrm{sin}\left(\alpha \right){e}_{1}{e}_{2}$

is a group homomorphism from the real numbers onto the circle group. The twisted adjoint representation ${\rho }_{2}$ maps the element $\mathrm{cos}\left(\alpha \right)+\mathrm{sin}\left(\alpha \right){e}_{1}{e}_{2}\in \mathrm{Spin}\left(2\right)$ to the transformation

${x}_{1}{e}_{1}+{x}_{2}{e}_{2}↦\left(\mathrm{cos}\left(\alpha \right)+\mathrm{sin}\left(\alpha \right){e}_{1}{e}_{2}\right)\left({x}_{1}{e}_{1}+{x}_{2}{e}_{2}\right)\left(\mathrm{cos}\left(\alpha \right)-\mathrm{sin}\left(\alpha \right){e}_{1}{e}_{2}\right)$$={x}_{1}\left(\mathrm{cos}{\left(\alpha \right)}^{2}-\mathrm{sin}{\left(\alpha \right)}^{2}\right){e}_{1}-2{x}_{2}\mathrm{cos}\left(\alpha \right)\mathrm{sin}\left(\alpha \right){e}_{1}$$+2{x}_{1}\mathrm{cos}\left(\alpha \right)\mathrm{sin}\left(\alpha \right){e}_{2}+{x}_{2}\left(\mathrm{cos}{\left(\alpha \right)}^{2}-\mathrm{sin}{\left(\alpha \right)}^{2}\right){e}_{2}$$=\left({x}_{1}\mathrm{cos}\left(2\alpha \right)-{x}_{2}\mathrm{sin}\left(2\alpha \right)\right){e}_{1}+\left({x}_{1}\mathrm{sin}\left(2\alpha \right)+{x}_{2}\mathrm{cos}\left(2\alpha \right)\right){e}_{2},$

i.e., the rotation of the Euclidean plane ${ℝ}^{2}$ by the angle $2\alpha$, corresponding to the matrix

$\left(\begin{array}{cc}\hfill \mathrm{cos}\left(2\alpha \right)\hfill & \hfill -\mathrm{sin}\left(2\alpha \right)\hfill \\ \hfill \mathrm{sin}\left(2\alpha \right)\hfill & \hfill \mathrm{cos}\left(2\alpha \right)\hfill \end{array}\right)\in \mathrm{SO}\left(2\right).$

In other words, ${\rho }_{2}$ is the double cover of the circle group by itself, cf. Theorem 8 (b).

Similarly, each element $g\in \mathrm{Spin}\left(3\right)$ is of the form

$g=\prod _{i=1}^{2k}\left({a}_{i}{e}_{1}+{b}_{i}{e}_{2}+{c}_{i}{e}_{3}\right)=a+b{e}_{1}{e}_{2}+c{e}_{2}{e}_{3}+d{e}_{3}{e}_{1}$

and each element $h\in \mathrm{Spin}\left(4\right)$ of the form

$h=\prod _{i=1}^{2k}\left({a}_{i}{e}_{1}+{b}_{i}{e}_{2}+{c}_{i}{e}_{3}+{d}_{i}{e}_{4}\right)$$={h}_{1}+{h}_{2}{e}_{1}{e}_{2}+{h}_{3}{e}_{2}{e}_{3}+{h}_{4}{e}_{3}{e}_{1}+{h}_{5}{e}_{1}{e}_{2}{e}_{3}{e}_{4}$$+{h}_{6}{e}_{4}{e}_{3}+{h}_{7}{e}_{4}{e}_{1}+{h}_{8}{e}_{4}{e}_{2}.$

7 The isomorphism $\mathrm{Spin}\left(4\right)\cong \mathrm{Spin}\left(3\right)×\mathrm{Spin}\left(3\right)$

In this section we recall special isomorphisms admitted by the groups $\mathrm{Spin}\left(3\right)$ and $\mathrm{Spin}\left(4\right)$. This structural information will only become relevant in Part III (Sections 12 and 13) of this article.

Denote by

$ℍ:=\left\{a+bi+cj+dk\mid a,b,c,d\in ℝ\right\}$

the real quaternions, identify $ℝ$ with the centre of $ℍ$ via $ℝ\to ℍ$, $a↦a$, let

$\overline{\cdot }:ℍ\to ℍ,x=a+bi+cj+dk↦\overline{x}=a-bi-cj-dk$

be the standard involution, and let

${\mathrm{U}}_{1}\left(ℍ\right):=\left\{x\in ℍ\mid x\overline{x}={1}_{ℍ}\right\}$

be the group of unit quaternions.

By [13, Section 1.4] one has

$\mathrm{Spin}\left(3\right)\cong {\mathrm{U}}_{1}\left(ℍ\right)\mathit{ }\text{and}\mathit{ }\mathrm{Spin}\left(4\right)\cong \mathrm{Spin}\left(3\right)×\mathrm{Spin}\left(3\right)\cong {\mathrm{U}}_{1}\left(ℍ\right)×{\mathrm{U}}_{1}\left(ℍ\right).$

The isomorphism $\mathrm{Spin}\left(3\right)\cong {\mathrm{U}}_{1}\left(ℍ\right)$ in fact is an immediate consequence of the isomorphism ${\mathrm{Cl}}^{0}\left(3\right)\cong ℍ$ from Remark 5 (b) plus the observation that this isomorphism transforms the spinor norm into the norm of the quaternions.

A canonical isomorphism $\mathrm{Spin}\left(4\right)\cong \mathrm{Spin}\left(3\right)×\mathrm{Spin}\left(3\right)\cong {\mathrm{U}}_{1}\left(ℍ\right)×{\mathrm{U}}_{1}\left(ℍ\right)$ can be described as follows (see [13, Section 1.4]). By Remark 14 each element of $\mathrm{Spin}\left(4\right)$ is of the form

$a+b{e}_{1}{e}_{2}+c{e}_{2}{e}_{3}+d{e}_{3}{e}_{1}+{a}^{\prime }{e}_{1}{e}_{2}{e}_{3}{e}_{4}+{b}^{\prime }{e}_{4}{e}_{3}+{c}^{\prime }{e}_{4}{e}_{1}+{d}^{\prime }{e}_{4}{e}_{2}.$

For

$i:={e}_{1}{e}_{2},$$j:={e}_{2}{e}_{3},$$k\mathit{ }:={e}_{3}{e}_{1},$$𝕀:={e}_{1}{e}_{2}{e}_{3}{e}_{4},$${i}^{\prime }:={e}_{4}{e}_{3},$${j}^{\prime }:={e}_{4}{e}_{1},$${k}^{\prime }\mathit{ }:={e}_{4}{e}_{2}$

one has

$ij=k,$$jk=i,$$ki\mathit{ }=j,$$i𝕀=𝕀i={i}^{\prime },$$j𝕀=𝕀j={j}^{\prime },$$k𝕀\mathit{ }=𝕀k={k}^{\prime },$${i}^{2}={j}^{2}={k}^{2}=-1,$${𝕀}^{2}=1,$$\sigma \left(𝕀\right)\mathit{ }=𝕀,$

where $\sigma \left(𝕀\right)$ denotes the Clifford conjugate of $𝕀$, cf. Definition 1. We conclude that for every $x\in \mathrm{Spin}\left(4\right)$ there exist uniquely determined $u=a+bi+cj+dk$, $v={a}^{\prime }+{b}^{\prime }i+{c}^{\prime }j+{d}^{\prime }k\in ℍ$ such that

$x=u+𝕀v.$

One computes

$N\left(x\right)=N\left(u+𝕀v\right)=\left(u+𝕀v\right)\left(\overline{u}+𝕀\overline{v}\right)=u\overline{u}+v\overline{v}+𝕀\left(u\overline{v}+v\overline{u}\right),$

i.e.,

$N\left(x\right)=1⇔u\overline{u}+v\overline{v}=1\text{and}u\overline{v}+v\overline{u}=0.$

Hence, for $1=N\left(x\right)=N\left(u+𝕀v\right)$, one has

$N\left(u+v\right)=\left(u+v\right)\left(\overline{u}+\overline{v}\right)=1,$$N\left(u-v\right)=\left(u-v\right)\left(\overline{u}-\overline{v}\right)=1.$

That is, the map

$\mathrm{Spin}\left(4\right)\to \mathrm{Spin}\left(3\right)×\mathrm{Spin}\left(3\right),u+𝕀v↦\left(u+v,u-v\right)$

is a well-defined bijection and, since

$\left(u+𝕀v\right)\left({u}^{\prime }+𝕀{v}^{\prime }\right)=u{u}^{\prime }+v{v}^{\prime }+𝕀\left(u{v}^{\prime }+v{u}^{\prime }\right)$

and

$\left(u+v,u-v\right)\left({u}^{\prime }+{v}^{\prime },{u}^{\prime }-{v}^{\prime }\right)=\left(u{u}^{\prime }+v{v}^{\prime }+u{v}^{\prime }+v{u}^{\prime },u{u}^{\prime }+v{v}^{\prime }-\left(u{v}^{\prime }+v{u}^{\prime }\right)\right),$

it is in fact an isomorphism of groups.

Consequently, there exists a group epimorphism

$\stackrel{~}{\eta }:\mathrm{Spin}\left(4\right)\to \mathrm{Spin}\left(3\right),u+𝕀v↦u+v.$

Using this isomorphism

$\mathrm{Spin}\left(4\right)\cong \mathrm{Spin}\left(3\right)×\mathrm{Spin}\left(3\right)\cong {\mathrm{U}}_{1}\left(ℍ\right)×{\mathrm{U}}_{1}\left(ℍ\right)$

there exists a natural homomorphism

$\mathrm{Spin}\left(4\right)\to \mathrm{SO}\left(ℍ\right)\cong \mathrm{SO}\left(4\right),\left(a,b\right)↦\left(x↦ax{b}^{-1}\right).$

Note that the restrictions $\left(a,1\right)↦\left(x↦ax\right)$ and $\left(1,b\right)↦\left(x↦x{b}^{-1}\right)$ are both injections of $\mathrm{Spin}\left(3\right)\cong {\mathrm{U}}_{1}\left(ℍ\right)$ into $\mathrm{GL}\left(ℍ\right)\cong \left(ℍ\\left\{0\right\},\cdot \right)$, in fact into $\mathrm{SO}\left(ℍ\right)$, as the norm is multiplicative. Since the kernel of this action has order two, the homomorphism $\mathrm{Spin}\left(4\right)\to \mathrm{SO}\left(ℍ\right)\cong \mathrm{SO}\left(4\right)$ must be onto by Proposition 8. We conclude that the group $\mathrm{SO}\left(4\right)$ is isomorphic to the group consisting of the maps

$ℍ\to ℍ,x↦ax{b}^{-1}\mathit{ }\text{for}a,b\in {\mathrm{U}}_{1}\left(ℍ\right)\text{;}$

for an alternative proof see [37, Lemma 11.22].

A similar argument (or a direct computation using the twisted adjoint representation) shows that the natural homomorphism

$\mathrm{Spin}\left(3\right)\to \mathrm{SO}\left({〈i,j,k〉}_{ℝ}\right)\cong \mathrm{SO}\left(3\right),a↦\left(x↦ax{a}^{-1}\right)$

is an epimorphism and, thus, that the group $\mathrm{SO}\left(3\right)$ is isomorphic to the group consisting of the maps

$ℍ\to ℍ,x↦ax{a}^{-1}\mathit{ }\text{for}a\in {\mathrm{U}}_{1}\left(ℍ\right)\text{;}$

There also exists a group epimorphism

$\eta :\mathrm{SO}\left(4\right)\to \mathrm{SO}\left(3\right)$

induced by the map

$\mathrm{SO}\left(4\right)\cong \left\{ℍ\to ℍ,x↦ax{b}^{-1}\mid a,b\in {\mathrm{U}}_{1}\left(ℍ\right)$$\to \left\{ℍ\to ℍ,x↦ax{a}^{-1}\mid a\in {\mathrm{U}}_{1}\left(ℍ\right)\right\}\cong \mathrm{SO}\left(3\right)$$\left(x↦ax{b}^{-1}\right)↦\left(x↦ax{a}^{-1}\right).$

Altogether, one obtains the following commutative diagram:

8 Lifting automorphism from $\mathrm{SO}\left(n\right)$ to $\mathrm{Spin}\left(n\right)$

For $\mathrm{SO}\left(2\right)×\mathrm{SO}\left(2\right)=\left\{\left(a,b\right)\mid a,b\in \mathrm{SO}\left(2\right)\right\}$ let

${\iota }_{1}:\mathrm{SO}\left(2\right)\to \mathrm{SO}\left(2\right)×\mathrm{SO}\left(2\right),x↦\left(x,1\right),$

and

${\iota }_{2}:\mathrm{SO}\left(2\right)\to \mathrm{SO}\left(2\right)×\mathrm{SO}\left(2\right),x↦\left(1,x\right).$

Similarly, for $\mathrm{Spin}\left(2\right)×\mathrm{Spin}\left(2\right)=\left\{\left(a,b\right)\mid a,b\in \mathrm{Spin}\left(2\right)\right\}$ let

${\stackrel{~}{\iota }}_{1}:\mathrm{Spin}\left(2\right)\to \mathrm{Spin}\left(2\right)×\mathrm{Spin}\left(2\right),x↦\left(x,1\right),$

and

${\stackrel{~}{\iota }}_{2}:\mathrm{Spin}\left(2\right)\to \mathrm{Spin}\left(2\right)×\mathrm{Spin}\left(2\right),x↦\left(1,x\right).$

Moreover, define

${\rho }_{2}×{\rho }_{2}:\mathrm{Spin}\left(2\right)×\mathrm{Spin}\left(2\right)\to \mathrm{SO}\left(2\right)×\mathrm{SO}\left(2\right),\left(a,b\right)↦\left({\rho }_{2}\left(a\right),{\rho }_{2}\left(b\right)\right).$

Hence

$\left({\rho }_{2}×{\rho }_{2}\right)\circ {\stackrel{~}{\iota }}_{1}={\iota }_{1}\circ {\rho }_{2},\left({\rho }_{2}×{\rho }_{2}\right)\circ {\stackrel{~}{\iota }}_{2}={\iota }_{2}\circ {\rho }_{2}.$

Furthermore, let

$\pi :\mathrm{Spin}\left(2\right)×\mathrm{Spin}\left(2\right)\to \mathrm{Spin}\left(2\right)×\mathrm{Spin}\left(2\right)/〈\left(-1,-1\right)〉$

be the canonical projection. By the homomorphism theorem of groups the map ${\rho }_{2}×{\rho }_{2}$ factors through $\mathrm{Spin}\left(2\right)×\mathrm{Spin}\left(2\right)/〈\left(-1,-1\right)〉$ and induces the following commutative diagram:

For $\alpha \in ℝ$ let

$D\left(\alpha \right):=\left(\begin{array}{cc}\hfill \mathrm{cos}\left(\alpha \right)\hfill & \hfill -\mathrm{sin}\left(\alpha \right)\hfill \\ \hfill \mathrm{sin}\left(\alpha \right)\hfill & \hfill \mathrm{cos}\left(\alpha \right)\hfill \end{array}\right)\in \mathrm{SO}\left(2\right)$

and

$S\left(\alpha \right):=\mathrm{cos}\left(\alpha \right)+\mathrm{sin}\left(\alpha \right){e}_{1}{e}_{2}\in \mathrm{Spin}\left(2\right).$

Then $\mathrm{Spin}\left(2\right)=\left\{S\left(\alpha \right)\mid \alpha \in ℝ\right\}$ and $\mathrm{SO}\left(2\right)=\left\{D\left(\alpha \right)\mid \alpha \in ℝ\right\}$ and there is a continuous group isomorphism

$\psi :\mathrm{SO}\left(2\right)\to \mathrm{Spin}\left(2\right),D\left(\alpha \right)↦S\left(\alpha \right).$

By the computation in Remark 14 the epimorphism ${\rho }_{2}$ from Theorem 8 satisfies ${\rho }_{2}=\mathrm{sq}\circ {\psi }^{-1}$, i.e.

${\rho }_{2}:\mathrm{Spin}\left(2\right)\to \mathrm{SO}\left(2\right),S\left(\alpha \right)↦D\left(2\alpha \right).$

Given an automorphism $\gamma \mathrm{\in }\mathrm{Aut}\mathit{}\mathrm{\left(}\mathrm{SO}\mathit{}\mathrm{\left(}\mathrm{2}\mathrm{\right)}\mathrm{\right)}$, there is a unique automorphism $\stackrel{\mathrm{~}}{\gamma }\mathrm{\in }\mathrm{Aut}\mathit{}\mathrm{\left(}\mathrm{Spin}\mathit{}\mathrm{\left(}\mathrm{2}\mathrm{\right)}\mathrm{\right)}$ such that ${\rho }_{\mathrm{2}}\mathrm{\circ }\stackrel{\mathrm{~}}{\gamma }\mathrm{=}\gamma \mathrm{\circ }{\rho }_{\mathrm{2}}$. Moreover, γ is continuous if and only if $\stackrel{\mathrm{~}}{\gamma }$ is continuous.

Proof.

Define $\stackrel{~}{\gamma }:=\psi \circ \gamma \circ {\psi }^{-1}$. Then

${\rho }_{2}\circ \stackrel{~}{\gamma }=\left(\mathrm{sq}\circ {\psi }^{-1}\right)\circ \left(\psi \circ \gamma \circ {\psi }^{-1}\right)$$=\mathrm{sq}\circ \gamma \circ {\psi }^{-1}$$=\gamma \circ \mathrm{sq}\circ {\psi }^{-1}$$=\gamma \circ {\rho }_{2}.$

Uniqueness follows as $Aut\left(\mathrm{SO}\left(2\right)\right)\to Aut\left(\mathrm{Spin}\left(2\right)\right)$, $\gamma ↦\psi \circ \gamma \circ {\psi }^{-1}$ is an isomorphism. ∎

Given an automorphism $\gamma \mathrm{\in }\mathrm{Aut}\mathit{}\mathrm{\left(}\mathrm{SO}\mathit{}\mathrm{\left(}\mathrm{2}\mathrm{\right)}\mathrm{×}\mathrm{SO}\mathit{}\mathrm{\left(}\mathrm{2}\mathrm{\right)}\mathrm{\right)}$, there is a unique automorphism $\stackrel{\mathrm{~}}{\gamma }\mathrm{\in }\mathrm{Aut}\mathit{}\mathrm{\left(}\mathrm{Spin}\mathit{}\mathrm{\left(}\mathrm{2}\mathrm{\right)}\mathrm{×}\mathrm{Spin}\mathit{}\mathrm{\left(}\mathrm{2}\mathrm{\right)}\mathrm{\right)}$ such that

$\left({\rho }_{2}×{\rho }_{2}\right)\circ \stackrel{~}{\gamma }=\gamma \circ \left({\rho }_{2}×{\rho }_{2}\right).$

Proof.

Let $\stackrel{~}{\gamma }:=\psi \circ \gamma \circ {\psi }^{-1}$, where

$\psi :\mathrm{SO}\left(2\right)×\mathrm{SO}\left(2\right)\to \mathrm{Spin}\left(2\right)×\mathrm{Spin}\left(2\right),\left(D\left(\alpha \right),D\left(\beta \right)\right)↦\left(S\left(\alpha \right),S\left(\beta \right)\right),$

and observe that ${\rho }_{2}×{\rho }_{2}=\mathrm{sq}\circ {\psi }^{-1}$. The claim now follows as in the proof of Proposition 2. ∎

Let $n\mathrm{\ge }\mathrm{3}$. Given an automorphism $\gamma \mathrm{\in }\mathrm{Aut}\mathit{}\mathrm{\left(}\mathrm{SO}\mathit{}\mathrm{\left(}n\mathrm{\right)}\mathrm{\right)}$, there is a unique automorphism $\stackrel{\mathrm{~}}{\gamma }\mathrm{\in }\mathrm{Aut}\mathit{}\mathrm{\left(}\mathrm{Spin}\mathit{}\mathrm{\left(}n\mathrm{\right)}\mathrm{\right)}$ such that

${\rho }_{n}\circ \stackrel{~}{\gamma }=\gamma \circ {\rho }_{n}.$

Proof.

For $n\ge 3$, both $\mathrm{SO}\left(n\right)$ and $\mathrm{Spin}\left(n\right)$ are perfect, cf. [24, Corollary 6.56]. By Theorem 8 (b) the group $\mathrm{Spin}\left(n\right)$ is a central extension of $\mathrm{SO}\left(n\right)$. Since $\mathrm{Spin}\left(n\right)$ is simply connected (see, e.g., [13, Section 1.8], it is in fact the universal central extension of $\mathrm{SO}\left(n\right)$.

The universal property of universal central extensions (cf. for example [19, Section 1.4C]) yields the claim: Indeed, there are unique homomorphisms

$\stackrel{~}{\gamma },{\stackrel{~}{\gamma }}^{\prime }:\mathrm{Spin}\left(n\right)\to \mathrm{Spin}\left(n\right)$

such that

$\gamma \circ {\rho }_{n}={\rho }_{n}\circ \stackrel{~}{\gamma }\mathit{ }\text{and}\mathit{ }{\gamma }^{-1}\circ {\rho }_{n}={\rho }_{n}\circ {\stackrel{~}{\gamma }}^{\prime }.$

Hence

${\rho }_{n}\circ \stackrel{~}{\gamma }\circ {\stackrel{~}{\gamma }}^{\prime }=\gamma \circ {\rho }_{n}\circ {\stackrel{~}{\gamma }}^{\prime }=\gamma \circ {\gamma }^{-1}\circ {\rho }_{n}={\rho }_{n}$

and, similarly,

${\rho }_{n}\circ {\stackrel{~}{\gamma }}^{\prime }\circ \stackrel{~}{\gamma }={\rho }_{n}.$

The universal property therefore implies $\stackrel{~}{\gamma }\circ {\stackrel{~}{\gamma }}^{\prime }=id={\stackrel{~}{\gamma }}^{\prime }\circ \stackrel{~}{\gamma }$, i.e., $\stackrel{~}{\gamma }$ is an automorphism.

In fact, all automorphisms are continuous by van der Waerden’s Continuity Theorem, cf. [24, Theorem 5.64]. ∎

For the following proposition recall the definitions of ${\epsilon }_{ij}$ in Notation 2 and of ${\stackrel{~}{\epsilon }}_{ij}$ in Lemma 10.

Let $\varphi \mathrm{:}\mathrm{Spin}\mathit{}\mathrm{\left(}\mathrm{2}\mathrm{\right)}\mathrm{\to }\mathrm{Spin}\mathit{}\mathrm{\left(}n\mathrm{\right)}$ be a homomorphism such that

$\mathrm{ker}\left({\rho }_{n}\circ \varphi \right)=\left\{1,-1\right\}\mathit{ }\text{𝑎𝑛𝑑}\mathit{ }{\rho }_{n}\circ \varphi ={\epsilon }_{ij}\circ {\rho }_{2}$

for some $i\mathrm{\ne }j\mathrm{\in }I$. Then $\varphi \mathrm{=}{\stackrel{\mathrm{~}}{\epsilon }}_{i\mathit{}j}$.

Proof.

By Consequence 12 one has

$\varphi \left(\mathrm{Spin}\left(2\right)\right)\subseteq \left(\rho _{n}{}^{-1}\circ {\epsilon }_{ij}\circ {\rho }_{2}\right)\left(\mathrm{Spin}\left(2\right)\right)=\rho _{n}{}^{-1}\left({\epsilon }_{ij}\left(\mathrm{SO}\left(2\right)\right)\right)={\stackrel{~}{\epsilon }}_{ij}\left(\mathrm{Spin}\left(2\right)\right).$

By hypothesis, $\mathrm{ker}\varphi \subseteq \left\{1,-1\right\}$. If $-1\in \mathrm{ker}\varphi$, then

$1=\varphi \left(-1\right)=\varphi \left(S\left(\pi \right)\right)=\varphi {\left(S\left(\frac{\pi }{2}\right)\right)}^{2},$

i.e., we have $\varphi \left(S\left(\frac{\pi }{2}\right)\right)\in \left\{1,-1\right\}$, whence $S\left(\frac{\pi }{2}\right)\in \mathrm{ker}\left({\rho }_{n}\circ \varphi \right)$, a contradiction. Consequently, ϕ is a monomorphism.

Consider the following commuting diagram:

One has ${\rho }_{2}\circ \stackrel{~}{\epsilon }_{ij}{}^{-1}\circ \varphi ={\rho }_{2}=id\circ {\rho }_{2}$. Since ϕ is injective, the map $\stackrel{~}{\epsilon }_{ij}{}^{-1}\circ \varphi$ is an automorphism of $\mathrm{Spin}\left(2\right)$. Hence Proposition 2 implies $\stackrel{~}{\epsilon }_{ij}{}^{-1}\circ \varphi =id$. ∎

9 $\mathrm{SO}\left(2\right)$-amalgams of simply-laced type

In this section we discuss amalgamation results for compact real orthogonal groups. The results and exposition are similar to [3, 17]. The key difference is that the amalgams in the present article are constructed starting with the circle group $\mathrm{SO}\left(2\right)$ instead of the perfect group $\mathrm{SU}\left(2\right)$. This leads to some subtle complications that we will need to address below.

Recall the maps ${\epsilon }_{12},{\epsilon }_{23}:\mathrm{SO}\left(2\right)\to \mathrm{SO}\left(3\right)$ from Notation 2 and the maps ${\iota }_{1},{\iota }_{2}:\mathrm{SO}\left(2\right)\to \mathrm{SO}\left(2\right)×\mathrm{SO}\left(2\right)$ from Notation 1.

Let Π be a simply-laced diagram with labelling $\sigma :I\to V$. An $\mathrm{SO}\left(2\right)$-amalgam with respect to Π and σ is an amalgam

$\mathcal{𝒜}=\left\{{G}_{ij},{\varphi }_{ij}^{i}\mid i\ne j\in I\right\}$

such that

${G}_{ij}=\left\{\begin{array}{cc}\mathrm{SO}\left(3\right),\hfill & \text{if}{\left\{i,j\right\}}^{\sigma }\in E\left(\mathrm{\Pi }\right),\hfill \\ \mathrm{SO}\left(2\right)×\mathrm{SO}\left(2\right),\hfill & \text{if}{\left\{i,j\right\}}^{\sigma }\notin E\left(\mathrm{\Pi }\right),\hfill \end{array}\mathit{ }\text{for all}i\ne j\in I$

and for $i,

${\varphi }_{ij}^{i}\left(\mathrm{SO}\left(2\right)\right)=\left\{\begin{array}{cc}{\epsilon }_{12}\left(\mathrm{SO}\left(2\right)\right),\hfill & \text{if}{\left\{i,j\right\}}^{\sigma }\in E\left(\mathrm{\Pi }\right),\hfill \\ {\iota }_{1}\left(\mathrm{SO}\left(2\right)\right),\hfill & \text{if}{\left\{i,j\right\}}^{\sigma }\notin E\left(\mathrm{\Pi }\right),\hfill \end{array}$${\varphi }_{ij}^{j}\left(\mathrm{SO}\left(2\right)\right)=\left\{\begin{array}{cc}{\epsilon }_{23}\left(\mathrm{SO}\left(2\right)\right),\hfill & \text{if}{\left\{i,j\right\}}^{\sigma }\in E\left(\mathrm{\Pi }\right),\hfill \\ {\iota }_{2}\left(\mathrm{SO}\left(2\right)\right),\hfill & \text{if}{\left\{i,j\right\}}^{\sigma }\notin E\left(\mathrm{\Pi }\right).\hfill \end{array}$

The standard $\mathrm{SO}\left(2\right)$-amalgam with respect to Π and σ is the $\mathrm{SO}\left(2\right)$-amalgam

$\mathcal{𝒜}\left(\mathrm{\Pi },\sigma ,\mathrm{SO}\left(2\right)\right):=\left\{{G}_{ij},{\varphi }_{ij}^{i}\mid i\ne j\in I\right\}$

with respect to Π and σ with

${\varphi }_{ij}^{i}=\left\{\begin{array}{cc}{\epsilon }_{12},\hfill & \text{if}{\left\{i,j\right\}}^{\sigma }\in E\left(\mathrm{\Pi }\right),\hfill \\ {\iota }_{1},\hfill & \text{if}{\left\{i,j\right\}}^{\sigma }\notin E\left(\mathrm{\Pi }\right),\hfill \end{array}$${\varphi }_{ij}^{j}=\left\{\begin{array}{cc}{\epsilon }_{23},\hfill & \text{if}{\left\{i,j\right\}}^{\sigma }\in E\left(\mathrm{\Pi }\right),\hfill \\ {\iota }_{2},\hfill & \text{if}{\left\{i,j\right\}}^{\sigma }\notin E\left(\mathrm{\Pi }\right),\hfill \end{array}$

for all $i.

The key difference between the standard $\mathrm{SO}\left(2\right)$-amalgam and an arbitrary $\mathrm{SO}\left(2\right)$-amalgam $\mathcal{𝒜}=\left\{{G}_{ij},{\varphi }_{ij}^{i}\mid i\ne j\in I\right\}$ with respect to Π and σ is that, for instance, ${\epsilon }_{12}^{-1}\circ {\varphi }_{ij}^{i}$ can be an arbitrary automorphism of $\mathrm{SO}\left(2\right)$. Automatic continuity (like van der Waerden’s Continuity Theorem, cf. [24, Theorem 5.64 and Corollary 6.56]) fails for automorphisms of the circle group $\mathrm{SO}\left(2\right)$ whereas it does hold for the group $\mathrm{SO}\left(3\right)$. Hence, obviously, not every automorphism of $\mathrm{SO}\left(2\right)$ is induced by an automorphism of $\mathrm{SO}\left(3\right)$ and so it is generally not possible to undo the automorphism ${\epsilon }_{12}^{-1}\circ {\varphi }_{ij}^{i}$ inside $\mathrm{SO}\left(3\right)$. Therefore Goldschmidt’s Lemma (see [15, Lemma 2.7], also [25, Proposition 8.3.2], [14, Lemma 6.16]) implies that for each diagram Π there exist plenty of pairwise non-isomorphic abstract $\mathrm{SO}\left(2\right)$-amalgams.

However, by [27, Section 4.G], [21, Corollary 7.16], a split real Kac–Moody group and its maximal compact subgroup (i.e., the group of elements fixed by the Cartan–Chevalley involution) both carry natural group topologies that induce the Lie group topology on their respective fundamental subgroups of ranks one and two and make the respective embeddings continuous.

It is therefore meaningful to use continuous $\mathrm{SO}\left(2\right)$-amalgams for studying these maximal compact subgroups. Such continuous amalgams are uniquely determined by the underlying diagram Π, as we will see in Theorem 8 below.

For each group isomorphic to one of $\mathrm{SO}\left(2\right)$, $\mathrm{SO}\left(2\right)×\mathrm{SO}\left(2\right)$, $\mathrm{SO}\left(3\right)$, we fix a matrix representation that allows us to identify the respective groups accordingly. Our study of amalgams by Goldschmidt’s Lemma [15, Lemma 2.7] then reduces to the study of automorphisms of these groups.

Let

$D:=\left(\begin{array}{ccc}\hfill \hfill & \hfill \hfill & \hfill 1\hfill \\ \hfill \hfill & \hfill -1\hfill & \hfill \hfill \\ \hfill 1\hfill & \hfill \hfill & \hfill \hfill \end{array}\right)\in \mathrm{SO}\left(3\right).$

Then the map ${\gamma }_{D}\mathrm{:}\mathrm{SO}\mathit{}\mathrm{\left(}\mathrm{3}\mathrm{\right)}\mathrm{\to }\mathrm{SO}\mathit{}\mathrm{\left(}\mathrm{3}\mathrm{\right)}$, $A\mathrm{↦}D\mathrm{\cdot }A\mathrm{\cdot }{D}^{\mathrm{-}\mathrm{1}}\mathrm{=}D\mathrm{\cdot }A\mathrm{\cdot }D$ is an automorphism of $\mathrm{SO}\mathit{}\mathrm{\left(}\mathrm{3}\mathrm{\right)}$ such that

${\gamma }_{D}\circ {\epsilon }_{12}={\epsilon }_{23}\mathit{ }\text{𝑎𝑛𝑑}\mathit{ }{\gamma }_{D}\circ {\epsilon }_{23}={\epsilon }_{12}.$

Proof.

Given $\left(\begin{array}{cc}\hfill x\hfill & \hfill y\hfill \\ \hfill -y\hfill & \hfill x\hfill \end{array}\right)\in \mathrm{SO}\left(2\right)$, we have

$\left(\begin{array}{ccc}\hfill \hfill & \hfill \hfill & \hfill 1\hfill \\ \hfill \hfill & \hfill -1\hfill & \hfill \hfill \\ \hfill 1\hfill & \hfill \hfill & \hfill \hfill \end{array}\right)\cdot \left(\begin{array}{ccc}\hfill x\hfill & \hfill y\hfill & \hfill \hfill \\ \hfill -y\hfill & \hfill x\hfill & \hfill \hfill \\ \hfill \hfill & \hfill \hfill & \hfill 1\hfill \end{array}\right)\cdot \left(\begin{array}{ccc}\hfill \hfill & \hfill \hfill & \hfill 1\hfill \\ \hfill \hfill & \hfill -1\hfill & \hfill \hfill \\ \hfill 1\hfill & \hfill \hfill & \hfill \hfill \end{array}\right)=\left(\begin{array}{ccc}\hfill \hfill & \hfill \hfill & \hfill 1\hfill \\ \hfill y\hfill & \hfill -x\hfill & \hfill \hfill \\ \hfill x\hfill & \hfill y\hfill & \hfill \hfill \end{array}\right)\cdot \left(\begin{array}{ccc}\hfill \hfill & \hfill \hfill & \hfill 1\hfill \\ \hfill \hfill & \hfill -1\hfill & \hfill \hfill \\ \hfill 1\hfill & \hfill \hfill & \hfill \hfill \end{array}\right)$$=\left(\begin{array}{ccc}\hfill 1\hfill & \hfill \hfill & \hfill \hfill \\ \hfill \hfill & \hfill x\hfill & \hfill y\hfill \\ \hfill \hfill & \hfill -y\hfill & \hfill x\hfill \end{array}\right).$

The second assertion follows analogously. ∎

The only influence of the labelling σ of an amalgam is the choice of which of the vertices ${i}^{\sigma }$, ${j}^{\sigma }$ corresponds to which subgroup of ${G}_{ij}$. We now show that this choice does not affect the isomorphism type of the amalgam.

Let $\mathrm{\Pi }$ be a simply-laced diagram with labellings ${\sigma }_{\mathrm{1}}\mathrm{,}{\sigma }_{\mathrm{2}}\mathrm{:}I\mathrm{\to }V$. Then

$\mathcal{𝒜}\left(\mathrm{\Pi },{\sigma }_{1},\mathrm{SO}\left(2\right)\right)\cong \mathcal{𝒜}\left(\mathrm{\Pi },{\sigma }_{2},\mathrm{SO}\left(2\right)\right).$

Proof.

Denote $\mathcal{𝒜}:=\mathcal{𝒜}\left(\mathrm{\Pi },{\sigma }_{1},\mathrm{SO}\left(2\right)\right)$ and $\overline{\mathcal{𝒜}}:=\mathcal{𝒜}\left(\mathrm{\Pi },{\sigma }_{2},\mathrm{SO}\left(2\right)\right)$. Let $D\in \mathrm{SO}\left(3\right)$ be as in Lemma 4 and let $\pi :={\sigma }_{2}^{-1}\circ {\sigma }_{1}\in Sym\left(I\right)$. Notice that

${\overline{G}}_{\pi \left(i\right)\pi \left(j\right)}=\mathrm{SO}\left(3\right)⇔{\left\{\pi \left(i\right),\pi \left(j\right)\right\}}^{{\sigma }_{2}}\in E\left(\mathrm{\Pi }\right)$$⇔{\left\{i,j\right\}}^{\pi {\sigma }_{2}}\in E\left(\mathrm{\Pi }\right)$$⇔{\left\{i,j\right\}}^{{\sigma }_{1}{\sigma }_{2}^{-1}{\sigma }_{2}}\in E\left(\mathrm{\Pi }\right)$$⇔{\left\{i,j\right\}}^{{\sigma }_{1}}\in E\left(\mathrm{\Pi }\right)$$⇔{G}_{ij}=\mathrm{SO}\left(3\right).$

Given $i with ${\left\{i,j\right\}}^{{\sigma }_{1}}\in E\left(\mathrm{\Pi }\right)$, let

${\alpha }_{ij}:=\left\{\begin{array}{cc}{id}_{\mathrm{SO}\left(3\right)},\hfill & \text{if}\pi \left(i\right)<\pi \left(j\right),\hfill \\ {\gamma }_{D},\hfill & \text{if}\pi \left(i\right)>\pi \left(j\right),\hfill \end{array}$

and given $i with ${\left\{i,j\right\}}^{{\sigma }_{1}}\notin E\left(\mathrm{\Pi }\right)$, let

${\alpha }_{ij}:\mathrm{SO}\left(2\right)×\mathrm{SO}\left(2\right)\to \mathrm{SO}\left(2\right)×\mathrm{SO}\left(2\right),\left(x,y\right)↦\left\{\begin{array}{cc}\left(x,y\right),\hfill & \text{if}\pi \left(i\right)<\pi \left(j\right),\hfill \\ \left(y,x\right),\hfill & \text{if}\pi \left(i\right)>\pi \left(j\right).\hfill \end{array}$

Then the system $\alpha :=\left\{\pi ,{\alpha }_{ij}\mid i\ne j\in I\right\}:\mathcal{𝒜}\to \overline{\mathcal{𝒜}}$ is an isomorphism of amalgams. Indeed, given $i with ${\left\{i,j\right\}}^{{\sigma }_{1}}\in E\left(\mathrm{\Pi }\right)$, one has

${\alpha }_{ij}\circ {\varphi }_{ij}^{i}={\alpha }_{ij}\circ {\epsilon }_{12}=\left\{\begin{array}{cc}{id}_{\mathrm{SO}\left(3\right)}\circ {\epsilon }_{12}={\epsilon }_{12}={\overline{\varphi }}_{\pi \left(i\right)\pi \left(j\right)}^{\pi \left(i\right)},\hfill & \text{if}\pi \left(i\right)<\pi \left(j\right),\hfill \\ {\gamma }_{D}\circ {\epsilon }_{12}={\epsilon }_{23}={\overline{\varphi }}_{\pi \left(i\right)\pi \left(j\right)}^{\pi \left(j\right)},\hfill & \text{if}\pi \left(i\right)>\pi \left(j\right),\hfill \end{array}$

and

${\alpha }_{ij}\circ {\varphi }_{ij}^{j}={\alpha }_{ij}\circ {\epsilon }_{23}=\left\{\begin{array}{cc}{id}_{\mathrm{SO}\left(3\right)}\circ {\epsilon }_{23}={\epsilon }_{23}={\overline{\varphi }}_{\pi \left(i\right)\pi \left(j\right)}^{\pi \left(i\right)},\hfill & \text{if}\pi \left(i\right)<\pi \left(j\right),\hfill \\ {\gamma }_{D}\circ {\epsilon }_{23}={\epsilon }_{12}={\overline{\varphi }}_{\pi \left(i\right)\pi \left(j\right)}^{\pi \left(j\right)},\hfill & \text{if}\pi \left(i\right)>\pi \left(j\right).\hfill \end{array}$

The case $i with ${\left\{i,j\right\}}^{{\sigma }_{1}}\notin E\left(\mathrm{\Pi }\right)$ is verified similarly. ∎

As we have just seen, the labelling of a standard $\mathrm{SO}\left(2\right)$-amalgam is irrelevant for its isomorphism type. Hence, for a simply-laced diagram Π, we write $\mathcal{𝒜}\left(\mathrm{\Pi },\mathrm{SO}\left(2\right)\right)$ to denote this isomorphism type and, moreover, by slight abuse of notation to denote any representative $\mathcal{𝒜}\left(\mathrm{\Pi },\sigma ,\mathrm{SO}\left(2\right)\right)$ of this isomorphism type. It is called the standard $\mathrm{SO}\left(2\right)$-amalgam with respect to Π.

Let $B\mathrm{:=}\mathrm{\left(}\begin{array}{cc}\hfill \mathrm{1}\hfill & \hfill \hfill \\ \hfill \hfill & \hfill \mathrm{-}{I}_{\mathrm{2}}\hfill \end{array}\mathrm{\right)}$, $C\mathrm{:=}\mathrm{\left(}\begin{array}{cc}\hfill \mathrm{-}{I}_{\mathrm{2}}\hfill & \hfill \hfill \\ \hfill \hfill & \hfill \mathrm{1}\hfill \end{array}\mathrm{\right)}\mathrm{\in }\mathrm{SO}\mathit{}\mathrm{\left(}\mathrm{3}\mathrm{\right)}$. Then the following holds:

• (a)

The map ${\gamma }_{B}:\mathrm{SO}\left(3\right)\to \mathrm{SO}\left(3\right)$, $A↦B\cdot A\cdot {B}^{-1}$ is an automorphism of $\mathrm{SO}\left(3\right)$ such that

${\gamma }_{B}\circ {\epsilon }_{12}={\epsilon }_{12}\circ \mathrm{inv}\mathit{ }\text{𝑎𝑛𝑑}\mathit{ }{\gamma }_{B}\circ {\epsilon }_{23}={\epsilon }_{23}.$

• (b)

The map ${\gamma }_{C}:\mathrm{SO}\left(3\right)\to \mathrm{SO}\left(3\right)$, $A↦C\cdot A\cdot {C}^{-1}$ is an automorphism of $\mathrm{SO}\left(3\right)$ such that

${\gamma }_{C}\circ {\epsilon }_{12}={\epsilon }_{12}\mathit{ }\text{𝑎𝑛𝑑}\mathit{ }{\gamma }_{C}\circ {\epsilon }_{23}={\epsilon }_{23}\circ \mathrm{inv}.$

Proof.

(a) Given $A:=\left(\begin{array}{cc}\hfill x\hfill & \hfill y\hfill \\ \hfill -y\hfill & \hfill x\hfill \end{array}\right)\in \mathrm{SO}\left(2\right)$, we have

${\gamma }_{B}\left({\epsilon }_{12}\left(A\right)\right)=B\cdot \left(\begin{array}{ccc}\hfill x\hfill & \hfill y\hfill & \hfill \hfill \\ \hfill -y\hfill & \hfill x\hfill & \hfill \hfill \\ \hfill \hfill & \hfill \hfill & \hfill 1\hfill \end{array}\right)\cdot {B}^{-1}=\left(\begin{array}{ccc}\hfill x\hfill & \hfill -y\hfill & \hfill \hfill \\ \hfill y\hfill & \hfill x\hfill & \hfill \hfill \\ \hfill \hfill & \hfill \hfill & \hfill 1\hfill \end{array}\right)={\epsilon }_{12}\left({A}^{-1}\right).$

The second statement follows analogously.

Part (b) is shown with a similar computation. ∎

Let Π be a simply-laced diagram with labelling $\sigma \mathrm{:}I\mathrm{\to }V$ and let $\mathcal{A}\mathrm{=}\mathrm{\left\{}{G}_{i\mathit{}j}\mathrm{,}{\varphi }_{i\mathit{}j}^{i}\mathrm{\mid }i\mathrm{\ne }j\mathrm{\in }I\mathrm{\right\}}$ be a continuous $\mathrm{SO}\mathit{}\mathrm{\left(}\mathrm{2}\mathrm{\right)}$-amalgam with respect to Π and σ. Then $\mathcal{A}\mathrm{\cong }\mathcal{A}\mathit{}\mathrm{\left(}\mathrm{\Pi }\mathrm{,}\sigma \mathrm{,}\mathrm{SO}\mathit{}\mathrm{\left(}\mathrm{2}\mathrm{\right)}\mathrm{\right)}$.

Proof.

Denote $\overline{\mathcal{𝒜}}:=\mathcal{𝒜}\left(\mathrm{\Pi },\sigma ,\mathrm{SO}\left(2\right)\right)$. The only continuous automorphisms of the circle group $\mathrm{SO}\left(2\right)$ are $id$ and the inversion $\mathrm{inv}$. Since $\mathcal{𝒜}$ is continuous by hypothesis, for all $i with ${\left\{i,j\right\}}^{\sigma }\in E\left(\mathrm{\Pi }\right)$ we have

${\varphi }_{ij}^{i}\in \left\{{\epsilon }_{12},{\epsilon }_{12}\circ \mathrm{inv}\right\},{\varphi }_{ij}^{j}\in \left\{{\epsilon }_{23},{\epsilon }_{23}\circ \mathrm{inv}\right\}.$

Let $B,C\in \mathrm{SO}\left(3\right)$ be as in Lemma 7, let $\pi :={id}_{I}$, and given $i such that ${\left\{i,j\right\}}^{\sigma }\in E\left(\mathrm{\Pi }\right)$, let

${\alpha }_{ij}:=\left\{\begin{array}{cc}{id}_{\mathrm{SO}\left(3\right)},\hfill & \text{if}{\varphi }_{ij}^{i}={\epsilon }_{12},{\varphi }_{ij}^{j}={\epsilon }_{23},\hfill \\ {\gamma }_{B},\hfill & \text{if}{\varphi }_{ij}^{i}={\epsilon }_{12}\circ \mathrm{inv},{\varphi }_{ij}^{j}={\epsilon }_{23},\hfill \\ {\gamma }_{C},\hfill & \text{if}{\varphi }_{ij}^{i}={\epsilon }_{12},{\varphi }_{ij}^{j}={\epsilon }_{23}\circ \mathrm{inv},\hfill \\ {\gamma }_{B}\circ {\gamma }_{C},\hfill & \text{if}{\varphi }_{ij}^{i}={\epsilon }_{12}\circ \mathrm{inv},{\varphi }_{ij}^{j}={\epsilon }_{23}\circ \mathrm{inv}.\hfill \end{array}$

For $i with ${\left\{i,j\right\}}^{\sigma }\notin E\left(\mathrm{\Pi }\right)$, define

${\alpha }_{ij}:=\left({\iota }_{1}\circ {\left({\varphi }_{ij}^{i}\right)}^{-1}\right)×\left({\iota }_{2}\circ {\left({\varphi }_{ij}^{j}\right)}^{-1}\right).$

Then the system $\alpha :=\left\{\pi ,{\alpha }_{ij}\mid i\ne j\in I\right\}:\mathcal{𝒜}\to \overline{\mathcal{𝒜}}$ is an isomorphism of amalgams. ∎

The following is well known, e.g. [12, Theorem 1.2].

For $n\mathrm{\ge }\mathrm{3}$, the group $\mathrm{SO}\mathit{}\mathrm{\left(}n\mathrm{\right)}$ is a universal enveloping group of the amalgam $\mathcal{A}\mathit{}\mathrm{\left(}{A}_{n\mathrm{-}\mathrm{1}}\mathrm{,}\mathrm{SO}\mathit{}\mathrm{\left(}\mathrm{2}\mathrm{\right)}\mathrm{\right)}$.

Proof.

Let $I:=\left\{1,\mathrm{\dots },n-1\right\}$. The group $\mathrm{SO}\left(n\right)$ acts flag-transitively on the simply connected projective geometry $\mathcal{𝒢}:={ℙ}_{n-1}\left(ℝ\right)$; simple connectedness follows from [40, Theorem 13.32], [41, Theorem 2], [1, Proposition 11.1.9, Theorem 11.1.13], flag-transitivity of the action from the Iwasawa/QR-decomposition of ${\mathrm{SL}}_{n}\left(ℝ\right)$. A maximal flag is given by

${〈{e}_{1}〉}_{ℝ}\le {〈{e}_{1},{e}_{2}〉}_{ℝ}\le \mathrm{\cdots }\le {〈{e}_{1},\mathrm{\dots },{e}_{n-1}〉}_{ℝ}.$

Let T be the subgroup of $\mathrm{SO}\left(n\right)$ of diagonal matrices; it is isomorphic to ${C}_{2}^{n-1}$ (where ${C}_{2}$ is a cyclic group of order 2). For $1\le i\le n-1$, let ${H}_{i}\cong \mathrm{SO}\left(2\right)$ be the circle group acting naturally on ${〈{e}_{i},{e}_{i+1}〉}_{ℝ}$ and, for $1\le i\le n-2$, let ${H}_{i,i+1}\cong \mathrm{SO}\left(3\right)$ be the group acting naturally on ${〈{e}_{i},{e}_{i+1},{e}_{i+2}〉}_{ℝ}$. Furthermore, for all i and j with $1\le i, let ${H}_{ij}:={H}_{i}{H}_{j}\cong \mathrm{SO}\left(2\right)×\mathrm{SO}\left(2\right)$. Then the stabiliser of a sub-flag of co-rank one is of the form ${H}_{i}T$, $1\le i\le n-1$, and the stabiliser of a sub-flag of co-rank two is of the form ${H}_{ij}T$, $1\le i.

The group $T\cong {C}_{2}^{n-1}$ admits a presentation with all generators and relations contained in the rank two subgroups ${H}_{ij}$ of $\mathrm{SO}\left(n\right)$: Indeed, T is generated by the groups ${C}_{2}\cong {T}_{i}:=〈-1〉\le {H}_{i}\cong \mathrm{SO}\left(2\right)$ and for each $1\le i\ne j\le n-1$ the relation ${T}_{i}{T}_{j}={T}_{j}{T}_{i}$ is visible within ${H}_{ij}$. Therefore, by an iteration of Tits’s Lemma (see [42, Corollary 1], [25, Corollary 1.4.6]) with respect to the above maximal flag, the group $H:=\mathrm{SO}\left(n\right)$ is the universal enveloping group of the amalgam $\mathcal{𝒜}\left(\mathcal{𝒢},H\right)=\left\{{H}_{ij},{\mathrm{\Phi }}_{ij}^{i}\mid i\ne j\in I\right\}$, where ${\mathrm{\Phi }}_{ij}^{i}:{H}_{i}\to {H}_{ij}$ is the inclusion map for each $i\ne j\in I$. One has

${H}_{i}={\epsilon }_{\left\{i,i+1\right\}}\left(\mathrm{SO}\left(2\right)\right),i\in I,$

and for all $i,

${H}_{ij}=\left\{\begin{array}{cc}{\epsilon }_{\left\{i,i+1,i+2\right\}}\left(\mathrm{SO}\left(3\right)\right),\hfill & \text{if}j=i+1,\hfill \\ {\epsilon }_{\left\{i,i+1\right\}}\left(\mathrm{SO}\left(2\right)\right)×{\epsilon }_{\left\{j,j+1\right\}}\left(\mathrm{SO}\left(2\right)\right),\hfill & \text{if}j\ne i+1.\hfill \end{array}$

As a consequence, the system

$\alpha =\left\{{id}_{I},{\alpha }_{ij},{\alpha }_{i}\mid i\ne j\in I\right\}:\mathcal{𝒜}\left({A}_{n-1},\mathrm{SO}\left(2\right)\right)\to \mathcal{𝒜}\left(\mathcal{𝒢},H\right)$

with

${\alpha }_{i}={\epsilon }_{\left\{i,i+1\right\}}:\mathrm{SO}\left(2\right)\to {H}_{i},i\in I,$

and for all $i,

${\alpha }_{ij}=\left\{\begin{array}{cc}{\epsilon }_{\left\{i,i+1,i+2\right\}},\hfill & \text{if}j=i+1,\hfill \\ {\epsilon }_{\left\{i,i+1\right\}}×{\epsilon }_{\left\{j,j+1\right\}},\hfill & \text{if}j\ne i+1,\hfill \end{array}$

is an isomorphism of amalgams. ∎

The above proof mainly relies on geometric arguments in the Tits building of type ${A}_{n-1}$. We exploit this to generalise the above statements to other diagrams, see Theorems 2 and 15. The crucial observation to make is that – via the local-to-global principle – it basically suffices to understand the rank two situation in order to understand arbitrary types.

10 $\mathrm{Spin}\left(2\right)$-amalgams of simply-laced type

In analogy to Section 9 we now study the amalgamation of groups isomorphic to $\mathrm{Spin}\left(3\right)$, continuously glued to one another along circle groups. In particular, we describe $\mathrm{Spin}\left(n\right)$ as the universal enveloping group of its $\mathrm{Spin}\left(2\right)$-amalgam and relate the classification of continuous $\mathrm{Spin}\left(2\right)$-amalgams to the classification of continuous $\mathrm{SO}\left(2\right)$-amalgams via the lifting of automorphisms.

Recall the maps ${\stackrel{~}{\epsilon }}_{12},{\stackrel{~}{\epsilon }}_{23}:\mathrm{Spin}\left(2\right)\to \mathrm{Spin}\left(3\right)$ from Lemma 10 and the maps ${\stackrel{~}{\iota }}_{1},{\stackrel{~}{\iota }}_{2}:\mathrm{Spin}\left(2\right)\to \mathrm{Spin}\left(2\right)×\mathrm{Spin}\left(2\right)$ from Notation 1.

Let Π be a simply-laced diagram with labelling $\sigma :I\to V$. A $\mathrm{Spin}\left(2\right)$-amalgam with respect to Π and σ is an amalgam

$\mathcal{𝒜}=\left\{{G}_{ij},{\varphi }_{ij}^{i}\mid i\ne j\in I\right\}$

such that for all $i\ne j\in I$,

${G}_{ij}=\left\{\begin{array}{cc}\mathrm{Spin}\left(3\right),\hfill & \text{if}{\left\{i,j\right\}}^{\sigma }\in E\left(\mathrm{\Pi }\right),\hfill \\ \mathrm{Spin}\left(2\right)×\mathrm{Spin}\left(2\right),\hfill & \text{if}{\left\{i,j\right\}}^{\sigma }\notin E\left(\mathrm{\Pi }\right),\hfill \end{array}$

and for $i,

${\varphi }_{ij}^{i}\left(\mathrm{Spin}\left(2\right)\right)=\left\{\begin{array}{cc}{\stackrel{~}{\epsilon }}_{12}\left(\mathrm{Spin}\left(2\right)\right),\hfill & \text{if}{\left\{i,j\right\}}^{\sigma }\in E\left(\mathrm{\Pi }\right),\hfill \\ {\iota }_{1}\left(\mathrm{Spin}\left(2\right)\right),\hfill & \text{if}{\left\{i,j\right\}}^{\sigma }\notin E\left(\mathrm{\Pi }\right),\hfill \end{array}$${\varphi }_{ij}^{j}\left(\mathrm{Spin}\left(2\right)\right)=\left\{\begin{array}{cc}{\stackrel{~}{\epsilon }}_{23}\left(\mathrm{Spin}\left(2\right)\right),\hfill & \text{if}{\left\{i,j\right\}}^{\sigma }\in E\left(\mathrm{\Pi }\right),\hfill \\ {\iota }_{2}\left(\mathrm{Spin}\left(2\right)\right),\hfill & \text{if}{\left\{i,j\right\}}^{\sigma }\notin E\left(\mathrm{\Pi }\right).\hfill \end{array}$

The standard $\mathrm{Spin}\left(2\right)$-amalgam with respect to Π and σ is the (continuous) $\mathrm{Spin}\left(2\right)$-amalgam

$\mathcal{𝒜}\left(\mathrm{\Pi },\sigma ,\mathrm{Spin}\left(2\right)\right):=\left\{{G}_{ij},{\varphi }_{ij}^{i}\mid i\ne j\in I\right\}$

with respect to Π and σ with

${\varphi }_{ij}^{i}=\left\{\begin{array}{cc}{\stackrel{~}{\epsilon }}_{12},\hfill & \text{if}{\left\{i,j\right\}}^{\sigma }\in E\left(\mathrm{\Pi }\right),\hfill \\ {\stackrel{~}{\iota }}_{1},\hfill & \text{if}{\left\{i,j\right\}}^{\sigma }\notin E\left(\mathrm{\Pi }\right),\hfill \end{array}$${\varphi }_{ij}^{j}=\left\{\begin{array}{cc}{\stackrel{~}{\epsilon }}_{23},\hfill & \text{if}{\left\{i,j\right\}}^{\sigma }\in E\left(\mathrm{\Pi }\right),\hfill \\ {\stackrel{~}{\iota }}_{2},\hfill & \text{if}{\left\{i,j\right\}}^{\sigma }\notin E\left(\mathrm{\Pi }\right).\hfill \end{array}$

for $i.

Proposition 2 enables us to lift $\mathrm{SO}\left(2\right)$-amalgams to $\mathrm{Spin}\left(2\right)$-amalgams: Let Π be a simply-laced diagram with labelling $\sigma :I\to V$ and let

$\mathcal{𝒜}=\left\{{G}_{ij},{\varphi }_{ij}^{i}\mid i\ne j\in I\right\}$

be an $\mathrm{SO}\left(2\right)$-amalgam with respect to Π and σ. Given $i,j\in I$ with $i, there are ${\gamma }_{ij}^{i},{\gamma }_{ij}^{j}\in Aut\left(\mathrm{SO}\left(2\right)\right)$ such that

${\varphi }_{ij}^{i}=\left\{\begin{array}{cc}{\epsilon }_{12}\circ {\gamma }_{ij}^{i},\hfill & \text{if}{\left\{i,j\right\}}^{\sigma }\in E\left(\mathrm{\Pi }\right),\hfill \\ {\iota }_{1}\circ {\gamma }_{ij}^{i},\hfill & \text{if}{\left\{i,j\right\}}^{\sigma }\notin E\left(\mathrm{\Pi }\right),\hfill \end{array}$${\varphi }_{ij}^{j}=\left\{\begin{array}{cc}{\epsilon }_{23}\circ {\gamma }_{ij}^{j},\hfill & \text{if}{\left\{i,j\right\}}^{\sigma }\in E\left(\mathrm{\Pi }\right),\hfill \\ {\iota }_{2}\circ {\gamma }_{ij}^{j},\hfill & \text{if}{\left\{i,j\right\}}^{\sigma }\notin E\left(\mathrm{\Pi }\right).\hfill \end{array}$

We then lift ${\gamma }_{ij}^{i},{\gamma }_{ij}^{j}$ as in Lemma 2 to ${\stackrel{~}{\gamma }}_{ij}^{i},{\stackrel{~}{\gamma }}_{ij}^{j}\in Aut\left(\mathrm{Spin}\left(2\right)\right)$ and set

${\stackrel{~}{\varphi }}_{ij}^{i}:=\left\{\begin{array}{cc}{\stackrel{~}{\epsilon }}_{12}\circ {\stackrel{~}{\gamma }}_{ij}^{i},\hfill & \text{if}{\left\{i,j\right\}}^{\sigma }\in E\left(\mathrm{\Pi }\right),\hfill \\ {\stackrel{~}{\iota }}_{1}\circ {\stackrel{~}{\gamma }}_{ij}^{i},\hfill & \text{if}{\left\{i,j\right\}}^{\sigma }\notin E\left(\mathrm{\Pi }\right),\hfill \end{array}$${\stackrel{~}{\varphi }}_{ij}^{j}:=\left\{\begin{array}{cc}{\stackrel{~}{\epsilon }}_{23}\circ {\stackrel{~}{\gamma }}_{ij}^{j},\hfill & \text{if}{\left\{i,j\right\}}^{\sigma }\in E\left(\mathrm{\Pi }\right),\hfill \\ {\stackrel{~}{\iota }}_{2}\circ {\stackrel{~}{\gamma }}_{ij}^{j},\hfill & \text{if}{\left\{i,j\right\}}^{\sigma }\notin E\left(\mathrm{\Pi }\right),\hfill \end{array}$

and

${\stackrel{~}{G}}_{ij}:=\left\{\begin{array}{cc}\mathrm{Spin}\left(3\right),\hfill & \text{if}{\left\{i,j\right\}}^{\sigma }\in E\left(\mathrm{\Pi }\right),\hfill \\ \mathrm{Spin}\left(2\right)×\mathrm{Spin}\left(2\right),\hfill & \text{if}{\left\{i,j\right\}}^{\sigma }\notin E\left(\mathrm{\Pi }\right).\hfill \end{array}$

Let Π be a simply-laced diagram with labelling $\sigma :I\to V$ and let $\mathcal{𝒜}=\left\{{G}_{ij},{\varphi }_{ij}^{i}\mid i\ne j\in I\right\}$ be an $\mathrm{SO}\left(2\right)$-amalgam with respect to Π and σ. Then

$\stackrel{~}{\mathcal{𝒜}}:=\left\{{\stackrel{~}{G}}_{ij},{\stackrel{~}{\varphi }}_{ij}^{i}\mid i\ne j\in I\right\}$

is the induced $\mathrm{Spin}\left(2\right)$-amalgam with respect to Π and σ. We also set

${\rho }_{ij}:=\left\{\begin{array}{cc}{\rho }_{3},\hfill & \text{if}{\left\{i,j\right\}}^{\sigma }\in E\left(\mathrm{\Pi }\right),\hfill \\ {\rho }_{2}×{\rho }_{2},\hfill & \text{if}{\left\{i,j\right\}}^{\sigma }\notin E\left(\mathrm{\Pi }\right).\hfill \end{array}$

Let Π be a simply-laced diagram with labelling $\sigma \mathrm{:}I\mathrm{\to }V$, let $\mathcal{A}\mathrm{=}\mathrm{\left\{}{G}_{i\mathit{}j}\mathrm{,}{\varphi }_{i\mathit{}j}^{i}\mathrm{\mid }i\mathrm{\ne }j\mathrm{\in }I\mathrm{\right\}}$ be an $\mathrm{SO}\mathit{}\mathrm{\left(}\mathrm{2}\mathrm{\right)}$-amalgam with respect to Π and σ, and let ${\stackrel{\mathrm{~}}{\varphi }}_{i\mathit{}j}^{i}$ be as introduced in Notation 2. Then for all $i\mathrm{\ne }j\mathrm{\in }I$

${\varphi }_{ij}^{i}\circ {\rho }_{2}={\rho }_{ij}\circ {\stackrel{~}{\varphi }}_{ij}^{i}.$

Proof.

Without loss of generality suppose $i. Let ${\gamma }_{ij}^{i},{\gamma }_{ij}^{j}\in Aut\left(\mathrm{SO}\left(2\right)\right)$, ${\stackrel{~}{\gamma }}_{ij}^{i},{\stackrel{~}{\gamma }}_{ij}^{j}\in Aut\left(\mathrm{Spin}\left(2\right)\right)$, and let ${\stackrel{~}{\varphi }}_{ij}^{i}$ and ${\stackrel{~}{\varphi }}_{ij}^{j}$ be as introduced in Notation 2. Then, if ${\left\{i,j\right\}}^{\sigma }\in E\left(\mathrm{\Pi }\right)$, we find that

${\varphi }_{ij}^{i}\circ {\rho }_{2}={\epsilon }_{12}\circ {\gamma }_{ij}^{i}\circ {\rho }_{2}={\epsilon }_{12}\circ {\rho }_{2}\circ {\stackrel{~}{\gamma }}_{ij}^{i}$$={\rho }_{3}\circ {\epsilon }_{12}\circ {\stackrel{~}{\gamma }}_{ij}^{i}\mathit{ }\text{(by Remark 6.13)}$$={\rho }_{3}\circ {\stackrel{~}{\varphi }}_{ij}^{i}.$

Similarly we also conclude

${\varphi }_{ij}^{j}\circ {\rho }_{2}={\rho }_{3}\circ {\stackrel{~}{\varphi }}_{ij}^{j}.$

Moreover, in case ${\left\{i,j\right\}}^{\sigma }\notin E\left(\mathrm{\Pi }\right)$ we deduce

${\varphi }_{ij}^{i}\circ {\rho }_{2}={\iota }_{1}\circ {\gamma }_{ij}^{i}\circ {\rho }_{2}={\iota }_{1}\circ {\rho }_{2}\circ {\stackrel{~}{\gamma }}_{ij}^{i}$$=\left({\rho }_{2}×{\rho }_{2}\right)\circ {\stackrel{~}{\iota }}_{1}\circ {\stackrel{~}{\gamma }}_{ij}^{i}$$=\left({\rho }_{2}×{\rho }_{2}\right)\circ {\stackrel{~}{\varphi }}_{ij}^{i}.$

Again we conclude by a similar argument that also

${\varphi }_{ij}^{j}\circ {\rho }_{2}=\left({\rho }_{2}×{\rho }_{2}\right)\circ {\stackrel{~}{\varphi }}_{ij}^{j}.\mathit{∎}$

Clearly the construction of an induced $\mathrm{Spin}\left(2\right)$-amalgam is symmetric and can also be applied backwards: Starting with a $\mathrm{Spin}\left(2\right)$-amalgam $\stackrel{^}{\mathcal{𝒜}}$ one can construct an $\mathrm{SO}\left(2\right)$-amalgam $\mathcal{𝒜}$ such that $\stackrel{^}{\mathcal{𝒜}}=\stackrel{~}{\mathcal{𝒜}}$. In particular, we obtain an epimorphism for the standard $\mathrm{Spin}\left(2\right)$- and $\mathrm{SO}\left(2\right)$-amalgams with respect to Π and σ, which we denote by

${\pi }_{\mathrm{\Pi },\sigma }=\left\{{\mathrm{id}}_{I},{\rho }_{2},{\rho }_{ij}\right\}:\mathcal{𝒜}\left(\mathrm{\Pi },\sigma ,\mathrm{Spin}\left(2\right)\right)\to \mathcal{𝒜}\left(\mathrm{\Pi },\sigma ,\mathrm{SO}\left(2\right)\right).$

Let Π be a simply-laced diagram with labelling $\sigma \mathrm{:}I\mathrm{\to }V$, let ${\mathcal{A}}_{\mathrm{1}}$ and ${\mathcal{A}}_{\mathrm{2}}$ be $\mathrm{SO}\mathit{}\mathrm{\left(}\mathrm{2}\mathrm{\right)}$-amalgams with respect to Π and σ, and let

$\alpha =\left\{\pi ,{\alpha }_{ij}\mid i\ne j\in I\right\}:{\mathcal{𝒜}}_{1}\to {\mathcal{𝒜}}_{2}$

be an isomorphism of amalgams. Then there is a unique isomorphism

$\stackrel{~}{\alpha }=\left\{\pi ,{\stackrel{~}{\alpha }}_{ij}\mid i\ne j\in I\right\}:{\stackrel{~}{\mathcal{𝒜}}}_{1}\to {\stackrel{~}{\mathcal{𝒜}}}_{2}$

such that for all $i\mathrm{\ne }j\mathrm{\in }I$,

${\rho }_{\pi \left(i\right)\pi \left(j\right)}\circ {\stackrel{~}{\alpha }}_{ij}={\alpha }_{ij}\circ {\rho }_{ij}.$

Proof.

Suppose ${\mathcal{𝒜}}_{1}=\left\{{G}_{ij},{\varphi }_{ij}^{i}\mid i\ne j\in I\right\}$ and ${\mathcal{𝒜}}_{2}=\left\{{H}_{ij},{\psi }_{ij}^{i}\mid i\ne j\in I\right\}$. Let $i\ne j\in I$. Since both amalgams are defined with respect to Π and σ, and since ${G}_{ij}\cong {H}_{\pi \left(i\right)\pi \left(j\right)}$, we conclude that ${\left\{i,j\right\}}^{\sigma }$ is an edge if and only if ${\left\{i,j\right\}}^{\pi \sigma }$ is an edge. Hence up to relabelling and identifying ${G}_{ij}$ with its image under ${\alpha }_{ij}$, we may assume $\pi =id$ and ${G}_{ij}={H}_{ij}$ and, thus, ${\alpha }_{ij}\in Aut\left({G}_{ij}\right)$ with ${\alpha }_{ij}\circ {\varphi }_{ij}^{i}={\psi }_{ij}^{i}$. We distinguish two cases.

Case I: ${\mathrm{\left\{}𝐢\mathrm{,}𝐣\mathrm{\right\}}}^{𝛔}\mathrm{\in }𝐄\mathit{}\mathrm{\left(}\mathrm{\Pi }\mathrm{\right)}$. Then ${\stackrel{~}{G}}_{ij}=\mathrm{Spin}\left(3\right)$. Let ${\stackrel{~}{\alpha }}_{ij}\in Aut\left(\mathrm{Spin}\left(3\right)\right)$ be the unique automorphism from Proposition 4 satisfying ${\rho }_{3}\circ {\stackrel{~}{\alpha }}_{ij}={\alpha }_{ij}\circ {\rho }_{3}$. It remains to verify that this is compatible with the amalgam structure. Indeed,

${\rho }_{3}\circ {\stackrel{~}{\alpha }}_{ij}\circ {\stackrel{~}{\varphi }}_{ij}^{i}\circ {\left({\stackrel{~}{\psi }}_{ij}^{i}\right)}^{-1}={\alpha }_{ij}\circ {\rho }_{3}\circ {\stackrel{~}{\varphi }}_{ij}^{i}\circ {\left({\stackrel{~}{\psi }}_{ij}^{i}\right)}^{-1}$$={\alpha }_{ij}\circ {\varphi }_{ij}^{i}\circ {\rho }_{2}\circ {\left({\stackrel{~}{\psi }}_{ij}^{i}\right)}^{-1}$$\text{(by Lemma 10.4)}$$=\underset{={id}_{{\epsilon }_{12}\left(\mathrm{SO}\left(2\right)\right)}}{\underset{⏟}{{\alpha }_{ij}\circ {\varphi }_{ij}^{i}\circ {\left({\psi }_{ij}^{i}\right)}^{-1}}}\circ {\rho }_{3}$$\text{(by Lemma 10.4)}$$={\rho }_{3}\circ {id}_{{\stackrel{~}{\epsilon }}_{12}\left(\mathrm{Spin}\left(2\right)\right)}.$

Hence, by uniqueness in Proposition 4, one has

${\stackrel{~}{\alpha }}_{ij}\circ {\stackrel{~}{\varphi }}_{ij}^{i}\circ {\left({\stackrel{~}{\psi }}_{ij}^{i}\right)}^{-1}={id}_{{\stackrel{~}{\epsilon }}_{12}\left(\mathrm{Spin}\left(2\right)\right)},$

i.e., ${\stackrel{~}{\alpha }}_{ij}\circ {\stackrel{~}{\varphi }}_{ij}^{i}={\stackrel{~}{\psi }}_{ij}^{i}$.

Case II: ${\mathrm{\left\{}𝐢\mathrm{,}𝐣\mathrm{\right\}}}^{𝛔}\mathrm{\notin }𝐄\mathit{}\mathrm{\left(}\mathrm{\Pi }\mathrm{\right)}$. In this case, we have ${\stackrel{~}{G}}_{ij}=\mathrm{Spin}\left(2\right)×\mathrm{Spin}\left(2\right)$. Let ${\stackrel{~}{\alpha }}_{ij}\in Aut\left(\mathrm{Spin}\left(2\right)×\mathrm{Spin}\left(2\right)\right)$ be the unique automorphism from Corollary 3 satisfying $\left({\rho }_{2}×{\rho }_{2}\right)\circ {\stackrel{~}{\alpha }}_{ij}={\alpha }_{ij}\circ \left({\rho }_{2}×{\rho }_{2}\right)$. It remains to verify that this is compatible with the amalgam structure. Indeed,

$\left({\rho }_{2}×{\rho }_{2}\right)\circ {\stackrel{~}{\alpha }}_{ij}\circ {\stackrel{~}{\varphi }}_{ij}^{i}\circ {\left({\stackrel{~}{\psi }}_{ij}^{i}\right)}^{-1}$$={\alpha }_{ij}\circ \left({\rho }_{2}×{\rho }_{2}\right)\circ {\stackrel{~}{\varphi }}_{ij}^{i}\circ {\left({\stackrel{~}{\psi }}_{ij}^{i}\right)}^{-1}$$={\alpha }_{ij}\circ {\varphi }_{ij}^{i}\circ {\rho }_{2}\circ {\left({\stackrel{~}{\psi }}_{ij}^{i}\right)}^{-1}$$\text{(by Lemma 10.4)}$$=\underset{={id}_{{\iota }_{1}\left(\mathrm{SO}\left(2\right)\right)}}{\underset{⏟}{{\alpha }_{ij}\circ {\varphi }_{ij}^{i}\circ {\left({\psi }_{ij}^{i}\right)}^{-1}}}\circ \left({\rho }_{2}×{\rho }_{2}\right)$$\text{(by Lemma 10.4)}$$=\left({\rho }_{2}×{\rho }_{2}\right)\circ {id}_{{\stackrel{~}{\iota }}_{1}\left(\mathrm{Spin}\left(2\right)\right)}.$

By uniqueness in Corollary 3, one concludes as in the previous case that

${\stackrel{~}{\alpha }}_{ij}\circ {\stackrel{~}{\varphi }}_{ij}^{i}={\stackrel{~}{\psi }}_{ij}^{i}.\mathit{∎}$

Let Π be a simply-laced diagram with labellings ${\sigma }_{\mathrm{1}}\mathrm{,}{\sigma }_{\mathrm{2}}$. Then $\mathcal{A}\mathit{}\mathrm{\left(}\mathrm{\Pi }\mathrm{,}{\sigma }_{\mathrm{1}}\mathrm{,}\mathrm{Spin}\mathit{}\mathrm{\left(}\mathrm{2}\mathrm{\right)}\mathrm{\right)}\mathrm{\cong }\mathcal{A}\mathit{}\mathrm{\left(}\mathrm{\Pi }\mathrm{,}{\sigma }_{\mathrm{2}}\mathrm{,}\mathrm{Spin}\mathit{}\mathrm{\left(}\mathrm{2}\mathrm{\right)}\mathrm{\right)}$.

Proof.

Let ${\mathcal{𝒜}}_{1}:=\mathcal{𝒜}\left(\mathrm{\Pi },{\sigma }_{1},\mathrm{SO}\left(2\right)\right)$ and ${\mathcal{𝒜}}_{2}:=\mathcal{𝒜}\left(\mathrm{\Pi },{\sigma }_{2},\mathrm{SO}\left(2\right)\right)$. The definitions imply ${\stackrel{~}{\mathcal{𝒜}}}_{1}=\mathcal{𝒜}\left(\mathrm{\Pi },{\sigma }_{1},\mathrm{Spin}\left(2\right)\right)$ and ${\stackrel{~}{\mathcal{𝒜}}}_{2}=\mathcal{𝒜}\left(\mathrm{\Pi },{\sigma }_{2},\mathrm{Spin}\left(2\right)\right)$. Moreover, one has ${\mathcal{𝒜}}_{1}\cong {\mathcal{𝒜}}_{2}$ by Consequence 5. The claim of the corollary now follows by applying Proposition 6. ∎

As before, for a simply-laced diagram Π with labelling σ, we write $\mathcal{𝒜}\left(\mathrm{\Pi },\mathrm{Spin}\left(2\right)\right)$ to denote the isomorphism type of $\mathcal{𝒜}\left(\mathrm{\Pi },\sigma ,\mathrm{Spin}\left(2\right)\right)$ and, by slight abuse of notation, any representative of this isomorphism type. It is called the standard $\mathrm{Spin}\left(2\right)$-amalgam with respect to Π.

Let Π be a simply-laced diagram with labelling $\sigma \mathrm{:}I\mathrm{\to }V$ and let $\stackrel{\mathrm{~}}{\mathcal{A}}$ be a continuous $\mathrm{Spin}\mathit{}\mathrm{\left(}\mathrm{2}\mathrm{\right)}$-amalgam with respect to Π and σ. Then

$\stackrel{~}{\mathcal{𝒜}}\cong \mathcal{𝒜}\left(\mathrm{\Pi },\mathrm{Spin}\left(2\right)\right).$

Proof.

Let $\mathcal{𝒜}$ be the continuous $\mathrm{SO}\left(2\right)$-amalgam that induces $\stackrel{~}{\mathcal{𝒜}}$, which exists by Remark 5. By Theorem 8, one has $\mathcal{𝒜}\cong \mathcal{𝒜}\left(\mathrm{\Pi },\mathrm{SO}\left(2\right)\right)$. Proposition 6 yields the claim, since $\mathcal{𝒜}\left(\mathrm{\Pi },\mathrm{SO}\left(2\right)\right)$ induces $\mathcal{𝒜}\left(\mathrm{\Pi },\mathrm{Spin}\left(2\right)\right)$. ∎

For $n\mathrm{\ge }\mathrm{3}$, the group $\mathrm{Spin}\mathit{}\mathrm{\left(}n\mathrm{\right)}$ is the universal enveloping group of $\mathcal{A}\mathit{}\mathrm{\left(}{A}_{n\mathrm{-}\mathrm{1}}\mathrm{,}\mathrm{Spin}\mathit{}\mathrm{\left(}\mathrm{2}\mathrm{\right)}\mathrm{\right)}$.

Proof.

The proof runs along the same lines as the proof of Theorem 9. Let $I:=\left\{1,\mathrm{\dots },n-1\right\}$. The group $\mathrm{Spin}\left(n\right)$ acts flag-transitively via the twisted adjoint representation (cf. Theorem 8 (b)) on the simply connected projective geometry $\mathcal{𝒢}:={ℙ}_{n-1}\left(ℝ\right)$ with fundamental maximal flag

${〈{e}_{1}〉}_{ℝ}\le {〈{e}_{1},{e}_{2}〉}_{ℝ}\le \mathrm{\cdots }\le {〈{e}_{1},\mathrm{\dots },{e}_{n-1}〉}_{ℝ}.$

By an iteration of Tits’s Lemma (see [42, Corollary 1] and [25, Corollary 1.4.6]) with respect to the above maximal flag, the group $H:=\mathrm{Spin}\left(n\right)$ is the universal enveloping group of the amalgam

$\mathcal{𝒜}\left(\mathcal{𝒢},H\right)=\left\{{H}_{ij},{\varphi }_{ij}^{i}\mid i\ne j\in I\right\},$

where the ${H}_{ij}$ are the “block-diagonal” rank two subgroups and ${\mathrm{\Phi }}_{ij}^{i}:{H}_{i}\to {H}_{ij}$ is the inclusion map for each $i\ne j\in I$. By Consequence 12 and Remark 13, one has

${H}_{i}={\stackrel{~}{\epsilon }}_{\left\{i,i+1\right\}}\left(\mathrm{Spin}\left(2\right)\right),i\in I,$

and for all $i,

${H}_{ij}=\left\{\begin{array}{cc}{\stackrel{~}{\epsilon }}_{\left\{i,i+1,i+2\right\}}\left(\mathrm{Spin}\left(3\right)\right),\hfill & \text{if}j=i+1,\hfill \\ {\stackrel{~}{\epsilon }}_{\left\{i,i+1\right\}}\left(\mathrm{Spin}\left(2\right)\right)\cdot {\stackrel{~}{\epsilon }}_{\left\{j,j+1\right\}}\left(\mathrm{Spin}\left(2\right)\right),\hfill & \text{if}j\ne i+1.\hfill \end{array}$

As a consequence, the system

$\alpha =\left\{{id}_{I},{\alpha }_{ij},{\alpha }_{i}\mid i\ne j\in I\right\}:\mathcal{𝒜}\left({A}_{n-1},\mathrm{Spin}\left(2\right)\right)\to \mathcal{𝒜}\left(\mathcal{𝒢},H\right)$

with

${\alpha }_{i}={\stackrel{~}{\epsilon }}_{\left\{i,i+1\right\}}:\mathrm{Spin}\left(2\right)\to {H}_{i},i\in I,$

and for all $i,

${\alpha }_{ij}=\left\{\begin{array}{cc}{\stackrel{~}{\epsilon }}_{\left\{i,i+1,i+2\right\}},\hfill & \text{if}j=i+1,\hfill \\ {\stackrel{~}{\epsilon }}_{\left\{i,i+1\right\}}\cdot {\stackrel{~}{\epsilon }}_{\left\{j,j+1\right\}},\hfill & \text{if}j\ne i+1,\hfill \end{array}$

is an epimorphism of $\mathrm{Spin}\left(2\right)$-amalgams. In fact, each ${\alpha }_{i}$ and each ${\alpha }_{ii+1}$ is an isomorphism, only the ${\alpha }_{ij}:\mathrm{Spin}\left(2\right)×\mathrm{Spin}\left(2\right)\to \mathrm{Spin}\left(2\right).\mathrm{Spin}\left(2\right)$, $j\ne i+1$, have a kernel of order two: the -1 in the left-hand factor gets identified with the -1 in the right-hand factor. One concludes that $\mathrm{Spin}\left(n\right)$ is an enveloping group of the amalgam $\mathcal{𝒜}\left({A}_{n-1},\mathrm{Spin}\left(2\right)\right)$. It remains to prove universality.

Let $\left(G,{\tau }_{ij}\right)$ be an arbitrary enveloping group of $\mathcal{𝒜}\left({A}_{n-1},\mathrm{Spin}\left(2\right)\right)$ and let i and j be given with $1\le i with $j\ne i+1$. By definition the following diagram commutes for $1\le a\ne b\ne c\le n-1$:

In particular, one has

$\left({\tau }_{i,i+2}\circ {\varphi }_{i,i+2}^{i}\right)\left(-1\right)$$=\left({\tau }_{i,i+1}\circ {\varphi }_{i,i+1}^{i}\right)\left(-1\right)\text{(set}b=i\text{,}a=i+1\text{,}c=i+2\text{)}$$=\left({\tau }_{i,i+1}\circ {\varphi }_{i,i+1}^{i+1}\right)\left(-1\right)\text{(since}{\stackrel{~}{\epsilon }}_{12}\left(-1\right)={\stackrel{~}{\epsilon }}_{23}\left(-1\right)\text{)}$$=\left({\tau }_{i+1,i+2}\circ {\varphi }_{i+1,i+2}^{i+1}\right)\left(-1\right)\text{(set}b=i+1\text{,}a=i\text{,}c=i+2\text{)}$$=\left({\tau }_{i+1,i+2}\circ {\varphi }_{i+1,i+2}^{i+2}\right)\left(-1\right)\text{(since}{\stackrel{~}{\epsilon }}_{12}\left(-1\right)={\stackrel{~}{\epsilon }}_{23}\left(-1\right)\text{)}$$=\left({\tau }_{i,i+2}\circ {\varphi }_{i,i+2}^{i+2}\right)\left(-1\right)\text{(set}b=i+2\text{,}a=i\text{,}c=i+1\text{)}.$

We conclude by induction that ${\tau }_{ij}:\mathrm{Spin}\left(2\right)×\mathrm{Spin}\left(2\right)\to G$, $j\ne i+1$, always factors through $\mathrm{Spin}\left(2\right).\mathrm{Spin}\left(2\right)$ or, in other words, $\tau :\mathcal{𝒜}\left({A}_{n-1},\mathrm{Spin}\left(2\right)\right)\to G$ always factors through $\mathcal{𝒜}\left(\mathcal{𝒢},H\right)$. That is, the universal enveloping group $\mathrm{Spin}\left(n\right)$ of $\mathcal{𝒜}\left(\mathcal{𝒢},H\right)$ is also a universal enveloping group of $\mathcal{𝒜}\left({A}_{n-1},\mathrm{Spin}\left(2\right)\right)$. ∎

The proof of Theorem 10 would become a bit easier if one replaced $\mathrm{Spin}\left(2\right)×\mathrm{Spin}\left(2\right)$ by $\mathrm{Spin}\left(2\right).\mathrm{Spin}\left(2\right)$ in Definition 1. The setup we chose, on the other hand, makes it easier to deal with reducible diagrams. Of course, one could a priori try to just restrict oneself to the case of irreducible diagrams, which in the case of simply-laced diagrams is unproblematic. However, when dealing with arbitrary diagrams it will turn out that it is more natural to also allow reducible diagrams.

11 Spin covers of simply-laced type

Let $\mathrm{\Pi }=\left(V,E\right)$ be a (finite) simply-laced diagram with labelling $\sigma :I\to V$ and let $c\left(\mathrm{\Pi }\right)$ denote the number of connected components of Π. A component labelling of Π is a map $\mathcal{𝒦}:V\to \left\{1,\mathrm{\dots },c\left(\mathrm{\Pi }\right)\right\}$ such that $u,v\in V$ are in the same connected component of Π if and only if $\mathcal{𝒦}\left(u\right)=\mathcal{𝒦}\left(v\right)$.

Throughout this section, let Π be a (finite) simply-laced diagram, $\sigma :I\to V$ a labelling and $\mathcal{𝒦}$ a component labelling.

Generalizing Theorem 9, the universal enveloping group of a continuous $\mathrm{SO}\left(2\right)$-amalgam over an arbitrary simply-laced diagram Π is isomorphic to the maximal compact subgroup of the corresponding split real Kac–Moody group (cf. Theorem 2). The goal of this section is to construct and investigate its spin cover, which will arise as the universal enveloping group of the continuous $\mathrm{Spin}\left(2\right)$-amalgam over the same simply-laced diagram Π. In the case of ${E}_{10}$ its existence has been conjectured by Damour and Hillmann in [9, Section 3.5, p. 24].

Additional key ingredients, next to transitive actions on buildings and the theory of $\mathrm{SO}\left(2\right)$- and $\mathrm{Spin}\left(2\right)$-amalgams developed so far, will be the generalised spin representations constructed in [20] and the Iwasawa decomposition of split real Kac–Moody groups studied, for example, in [12]. For definitions and details on Kac–Moody theory we refer the reader to [26], [34], [20, Section 1], [21, Section 7], [31, Chapter 5].

Let Π be a simply-laced diagram, let $G\mathit{}\mathrm{\left(}\mathrm{\Pi }\mathrm{\right)}$ be the corresponding simply connected split real Kac–Moody group, and let $K\mathit{}\mathrm{\left(}\mathrm{\Pi }\mathrm{\right)}$ be its maximal compact subgroup, i.e., the subgroup fixed by the Cartan–Chevalley involution. Then there exists a faithful universal enveloping morphism

${\tau }_{K\left(\mathrm{\Pi }\right)}:\mathcal{𝒜}\left(\mathrm{\Pi },\mathrm{SO}\left(2\right)\right)\to K\left(\mathrm{\Pi }\right).$

Proof.

For $i\in I$ denote by ${G}_{i}$ the fundamental rank one subgroups of $G\left(\mathrm{\Pi }\right)$ and, for $i\ne j\in I$, by ${G}_{ij}$ the fundamental rank two subgroups. The groups ${G}_{i}$ are isomorphic to ${\mathrm{SL}}_{2}\left(ℝ\right)$ and the groups ${G}_{ij}$ are isomorphic to ${\mathrm{SL}}_{3}\left(ℝ\right)$ or to ${\mathrm{SL}}_{2}\left(ℝ\right)×{\mathrm{SL}}_{2}\left(ℝ\right)$, depending on whether the vertices ${i}^{\sigma }$, ${j}^{\sigma }$ of Π are joined by an edge or not. The Cartan–Chevalley involution ω leaves the groups ${G}_{i}$, ${G}_{ij}$ setwise invariant and, in fact, induces the transpose-inverse map on these groups. Define ${H}_{i}:={Fix}_{{G}_{i}}\left(\omega \right)\cong \mathrm{SO}\left(2\right)$ and ${H}_{ij}:={Fix}_{{G}_{ij}}\left(\omega \right)$, the latter being isomorphic to $\mathrm{SO}\left(3\right)$ or to $\mathrm{SO}\left(2\right)×\mathrm{SO}\left(2\right)$, and let ${\psi }_{ij}^{i}:{H}_{i}\to {H}_{ij}$ denote the (continuous) inclusion map for $i\ne j\in I$. By [12, Theorem 1.2], the group $K\left(\mathrm{\Pi }\right)$ is the universal enveloping group of the amalgam ${\mathcal{𝒜}}_{1}:=\left\{{H}_{ij},{\psi }_{ij}^{i}\mid i\ne j\in I\right\}$.

Given $i such that ${\left\{i,j\right\}}^{\sigma }\in E\left(\mathrm{\Pi }\right)$, by Theorem 8 applied to the subdiagram of Π of type ${A}_{2}$ consisting of the vertices ${i}^{\sigma }$, ${j}^{\sigma }$ there is a continuous isomorphism ${\alpha }_{ij}:{H}_{ij}\to \mathrm{SO}\left(3\right)$ such that

$\left({\alpha }_{ij}\circ {\psi }_{ij}^{i}\right)\left({H}_{i}\right)={\epsilon }_{12}\left(\mathrm{SO}\left(2\right)\right),\left({\alpha }_{ij}\circ {\psi }_{ij}^{j}\right)\left({H}_{j}\right)={\epsilon }_{23}\left(\mathrm{SO}\left(2\right)\right).$

Let $i\in I$ and choose $j\in I$ such that ${\left\{i,j\right\}}^{\sigma }\in E\left(\mathrm{\Pi }\right)$. Define

${\alpha }_{i}:=\left\{\begin{array}{cc}{\epsilon }_{12}^{-1}\circ {\alpha }_{ij}\circ {\psi }_{ij}^{i}:{H}_{i}\to \mathrm{SO}\left(2\right),\hfill & \text{if}ij.\hfill \end{array}$

For $i such that ${\left\{i,j\right\}}^{\sigma }\notin E\left(\mathrm{\Pi }\right)$, define

${\alpha }_{ij}:={\alpha }_{i}×{\alpha }_{j}:{H}_{ij}={H}_{i}×{H}_{j}\to \mathrm{SO}\left(2\right)×\mathrm{SO}\left(2\right).$

For arbitrary $i\ne j\in I$, let

${K}_{ij}:=\left\{\begin{array}{cc}\mathrm{SO}\left(3\right),\hfill & \text{if}{\left\{i,j\right\}}^{\sigma }\in E\left(\mathrm{\Pi }\right),\hfill \\ \mathrm{SO}\left(2\right)×\mathrm{SO}\left(2\right),\hfill & \text{if}{\left\{i,j\right\}}^{\sigma }\notin E\left(\mathrm{\Pi }\right),\hfill \end{array}$

and

${\overline{\varphi }}_{ij}^{i}:={\alpha }_{ij}\circ {\psi }_{ij}^{i}\circ {\alpha }_{i}^{-1}:\mathrm{SO}\left(2\right)\to {K}_{ij}.$

Then ${\mathcal{𝒜}}_{2}:=\left\{{K}_{ij},{\overline{\varphi }}_{ij}^{i}\mid i\ne j\in I\right\}$ is an $\mathrm{SO}\left(2\right)$-amalgam with respect to Π and σ. Moreover, the system $\alpha =\left\{{id}_{I},{\alpha }_{ij},{\alpha }_{i}\mid i\ne j\in I\right\}:{\mathcal{𝒜}}_{1}\to {\mathcal{𝒜}}_{2}$ is an isomorphism of amalgams. Indeed, given $i\ne j\in I$, one has

${\overline{\varphi }}_{ij}^{i}\circ {\alpha }_{i}={\alpha }_{ij}\circ {\psi }_{ij}^{i}\circ {\alpha }_{i}^{-1}\circ {\alpha }_{i}={\alpha }_{ij}\circ {\psi }_{ij}^{i}.$

Finally, ${\alpha }_{i}$ is continuous for each $i\in I$, whence ${\overline{\varphi }}_{ij}^{i}$ is continuous for all $i\ne j\in I$. Therefore, ${\mathcal{𝒜}}_{2}$ is a continuous $\mathrm{SO}\left(2\right)$-amalgam with respect to Π and σ so that ${\mathcal{𝒜}}_{1}\cong {\mathcal{𝒜}}_{2}\cong \mathcal{𝒜}\left(\mathrm{\Pi },\mathrm{SO}\left(2\right)\right)$ by Theorem 8. ∎

For consistency, we fix the groups and connecting morphisms in the standard $\mathrm{SO}\left(2\right)$-amalgam with respect to Π as follows (cf. Definition 6):

$\mathcal{𝒜}\left(\mathrm{\Pi },\mathrm{SO}\left(2\right)\right)=\left\{{K}_{ij},{\varphi }_{ij}^{i}\mid i\ne j\in I\right\}.$

Similarly for the standard $\mathrm{Spin}\left(2\right)$-amalgam with respect to $\mathrm{\Pi }$ (cf. Definition 8):

$\mathcal{𝒜}\left(\mathrm{\Pi },\mathrm{Spin}\left(2\right)\right)=\left\{{\stackrel{~}{K}}_{ij},{\stackrel{~}{\varphi }}_{ij}^{i}\mid i\ne j\in I\right\}.$

We denote the epimorphism of amalgams from Remark 5 by

${\pi }_{\mathrm{\Pi }}:\mathcal{𝒜}\left(\mathrm{\Pi },\mathrm{Spin}\left(2\right)\right)\to \mathcal{𝒜}\left(\mathrm{\Pi },\mathrm{SO}\left(2\right)\right).$

As discussed in Remark 2 the amalgam $\mathcal{𝒜}\left(\mathrm{\Pi },\mathrm{SO}\left(2\right)\right)$ consists of compact Lie groups with continuous connecting homomorphisms. On the other hand, the group $K\left(\mathrm{\Pi }\right)$ naturally carries a Hausdorff group topology that is ${k}_{\omega }$: Indeed, $K\left(\mathrm{\Pi }\right)$ is the subgroup of the unitary form studied in [14, Section 6] fixed by complex conjugation and it is the subgroup of the real Kac–Moody group $G\left(\mathrm{\Pi }\right)$ studied in [21, Section 7] fixed by the Cartan–Chevalley involution; both ambient groups are ${k}_{\omega }$ (by [14, Theorem 6.12], resp. [21, Theorem 7.22]) and, hence, so is any subgroup fixed by a continuous involution (cf. [14, Proposition 4.2(b)]). Note that the ${k}_{\omega }$-group topologies on $K\left(\mathrm{\Pi }\right)$ induced from the real Kac–Moody group $G\left(\mathrm{\Pi }\right)$ and from the unitary form coincide, as both are induced from the ${k}_{\omega }$-group topology on the ambient complex Kac–Moody group (cf. [14, Theorem 6.3], resp. [21, Theorem 7.22]).

Furthermore, a straightforward adaptation of the proof of [14, Proposition 6.9] implies that this ${k}_{\omega }$-group topology is the finest group topology with respect to the enveloping homomorphisms ${\tau }_{ij}:{K}_{ij}\to K\left(\mathrm{\Pi }\right)$. In other words, the obvious analog of [14, Theorem 6.12] and [21, Theorem 7.22] holds for $\left(K\left(\mathrm{\Pi }\right),{\tau }_{K\left(\mathrm{\Pi }\right)}\right)$. In particular, to any enveloping morphism $\psi =\left({\psi }_{ij}\right):\mathcal{𝒜}\left(\mathrm{\Pi },\mathrm{SO}\left(2\right)\right)\to X$ into a Hausdorff topological group X with continuous homomorphisms ${\psi }_{ij}:{K}_{ij}\to X$ there exists a unique continuous homomorphism $\xi :K\left(\mathrm{\Pi }\right)\to X$ such that the following diagram commutes:

Theorems 9 and 10 state that the double cover $\mathrm{Spin}\left(n\right)$ of $\mathrm{SO}\left(n\right)$ is the universal enveloping group of the two-fold central extension $\mathcal{𝒜}\left({A}_{n-1},\mathrm{Spin}\left(2\right)\right)$ of the amalgam $\mathcal{𝒜}\left({A}_{n-1},\mathrm{SO}\left(2\right)\right)$ as defined in Proposition 9. In view of Theorem 2 it is therefore natural to introduce the following notion.

The spin group $\mathrm{Spin}\left(\mathrm{\Pi }\right)$ with respect to Π is the canonical universal enveloping group of the (continuous) amalgam $\mathcal{𝒜}\left(\mathrm{\Pi },\mathrm{Spin}\left(2\right)\right)$ with the canonical universal enveloping morphism

${\tau }_{\mathrm{Spin}\left(\mathrm{\Pi }\right)}=\left\{{\tau }_{ij}\mid i\ne j\in I\right\}:\mathcal{𝒜}\left(\mathrm{\Pi },\mathrm{Spin}\left(2\right)\right)\to \mathrm{Spin}\left(\mathrm{\Pi }\right).$

$K\left(\mathrm{\Pi }\right)$ is an enveloping group of the amalgam $\mathcal{A}\mathit{}\mathrm{\left(}\mathrm{\Pi }\mathrm{,}\mathrm{Spin}\mathit{}\mathrm{\left(}\mathrm{2}\mathrm{\right)}\mathrm{\right)}$. There exists a unique central extension ${\rho }_{\mathrm{\Pi }}\mathrm{:}\mathrm{Spin}\mathit{}\mathrm{\left(}\mathrm{\Pi }\mathrm{\right)}\mathrm{\to }K\mathit{}\mathrm{\left(}\mathrm{\Pi }\mathrm{\right)}$ that makes the following diagram commute (cf. Notation 8):

where

${\tau }_{K\left(\mathrm{\Pi }\right)}=\left\{{\psi }_{ij}:{K}_{ij}\to K\left(\mathrm{\Pi }\right)\mid i\ne j\in I\right\}:\mathcal{𝒜}\left(\mathrm{\Pi },\mathrm{SO}\left(2\right)\right)\to K\left(\mathrm{\Pi }\right)$

is the universal enveloping morphism whose existence is guaranteed by Theorem 2.

Proof.

As in Definition 3, for $i\ne j\in I$ let ${\rho }_{ij}:{\stackrel{~}{K}}_{ij}\to {K}_{ij}$ be the epimorphism ${\rho }_{3}$ if ${\left\{i,j\right\}}^{\sigma }$ is an edge, and ${\rho }_{2}×{\rho }_{2}$ otherwise. Then, by Lemma 7(a), the group $K\left(\mathrm{\Pi }\right)$ with the homomorphisms

${\xi }_{ij}:={\psi }_{ij}\circ {\rho }_{ij}:{\stackrel{~}{K}}_{ij}\to K\left(\mathrm{\Pi }\right)$

for all $i\ne j\in I$ is an enveloping group of $\mathcal{𝒜}\left(\mathrm{\Pi },\sigma ,\mathrm{Spin}\left(2\right)\right)$. By universality of ${\tau }_{\mathrm{Spin}\left(\mathrm{\Pi }\right)}:\mathcal{𝒜}\left(\mathrm{\Pi },\mathrm{Spin}\left(2\right)\right)\to \mathrm{Spin}\left(\mathrm{\Pi }\right)$, Lemma 7(b) provides a unique epimorphism ${\rho }_{\mathrm{\Pi }}:\mathrm{Spin}\left(\mathrm{\Pi }\right)\to K\left(\mathrm{\Pi }\right)$ with the properties as claimed. This epimorphism is a central extension by Proposition 9. ∎

The following is a straightforward generalization of the observation we made towards the end of the proof of Theorem 10.

Let $i\mathrm{\ne }j\mathrm{\in }I$ and $k\mathrm{\ne }\mathrm{\ell }\mathrm{\in }I$. If ${i}^{\sigma }$ and ${k}^{\sigma }$ are in the same connected component of Π, then

${\tau }_{ij}\left({\stackrel{~}{\varphi }}_{ij}^{i}\left(-{1}_{\mathrm{Spin}\left(2\right)}\right)\right)={\tau }_{k\mathrm{\ell }}\left({\stackrel{~}{\varphi }}_{k\mathrm{\ell }}^{k}\left(-{1}_{\mathrm{Spin}\left(2\right)}\right)\right).$

Proof.

As ${i}^{\sigma }$ and ${k}^{\sigma }$ are in the same connected component, there exists a sequence ${i}_{0}:=i,{i}_{1},\mathrm{\dots },{i}_{n}:=k\in I$ such that $\left\{{i}_{r}^{\sigma },{i}_{r+1}^{\sigma }\right\}$ are edges for $0\le r. Thus ${\stackrel{~}{K}}_{{i}_{r}{i}_{r+1}}=\mathrm{Spin}\left(3\right)$ and by the definition of $\mathrm{Spin}\left(\mathrm{\Pi }\right)$ as the canonical universal enveloping group of the amalgam $\mathcal{𝒜}\left(\mathrm{\Pi },\mathrm{Spin}\left(2\right)\right)$, we have

${\stackrel{~}{\varphi }}_{{i}_{r}{i}_{r+1}}^{{i}_{r}}\left(-{1}_{\mathrm{Spin}\left(2\right)}\right)=-{1}_{\mathrm{Spin}\left(3\right)}={\stackrel{~}{\varphi }}_{{i}_{r}{i}_{r+1}}^{{i}_{r+1}}\left(-{1}_{\mathrm{Spin}\left(2\right)}\right).$

Hence

${\tau }_{ij}\left({\stackrel{~}{\varphi }}_{ij}^{i}\left(-{1}_{\mathrm{Spin}\left(2\right)}\right)\right)={\tau }_{{i}_{0}{i}_{1}}\left({\stackrel{~}{\varphi }}_{{i}_{0}{i}_{1}}^{{i}_{0}}\left(-{1}_{\mathrm{Spin}\left(2\right)}\right)\right)$$={\tau }_{{i}_{0}{i}_{1}}\left({\stackrel{~}{\varphi }}_{{i}_{0}{i}_{1}}^{{i}_{1}}\left(-{1}_{\mathrm{Spin}\left(2\right)}\right)\right)$$\mathrm{ }\mathrm{⋮}$$={\tau }_{{i}_{n-1}{i}_{n}}\left({\stackrel{~}{\varphi }}_{{i}_{n-1}{i}_{n}}^{{i}_{n}}\left(-{1}_{\mathrm{Spin}\left(2\right)}\right)\right)$$={\tau }_{{i}_{n-1}k}\left({\stackrel{~}{\varphi }}_{{i}_{n-1}k}^{k}\left(-{1}_{\mathrm{Spin}\left(2\right)}\right)\right)$$={\tau }_{k\mathrm{\ell }}\left({\stackrel{~}{\varphi }}_{k\mathrm{\ell }}^{k}\left(-{1}_{\mathrm{Spin}\left(2\right)}\right)\right).$

where the first and last equality hold due to the definition of enveloping homomorphisms. ∎

Thus the following is well-defined.

For $i\ne j\in I$ define

$-{1}_{\mathrm{Spin}\left(\mathrm{\Pi }\right),\mathcal{𝒦}\left(i\right)}:={\tau }_{ij}\left({\stackrel{~}{\varphi }}_{ij}^{i}\left(-{1}_{\mathrm{Spin}\left(2\right)}\right)\right)$

and

$Z:=〈-{1}_{\mathrm{Spin}\left(\mathrm{\Pi }\right),1},\mathrm{\dots },-{1}_{\mathrm{Spin}\left(\mathrm{\Pi }\right),c\left(\mathrm{\Pi }\right)}〉\le \mathrm{Spin}\left(\mathrm{\Pi }\right).$

The following are true:

• (a)

Z is contained in the centre of $\mathrm{Spin}\left(\mathrm{\Pi }\right)$.

• (b)

$|Z|\le {2}^{c\left(\mathrm{\Pi }\right)}$.

Assertion (a) is immediate from Proposition 9 applied to the $\mathrm{Spin}\left(2\right)$-amalgam $\mathcal{𝒜}\left(\mathrm{\Pi },\mathrm{Spin}\left(2\right)\right)$ and the $\mathrm{SO}\left(2\right)$-amalgam $\mathcal{𝒜}\left(\mathrm{\Pi },\mathrm{SO}\left(2\right)\right)$ with $\stackrel{~}{U}=\mathrm{Spin}\left(2\right)$, $\stackrel{~}{V}=〈-1〉$ and $U=\mathrm{SO}\left(2\right)$. The second follows from the fact that Z is abelian by assertion (a) and admits a generating system of $c\left(\mathrm{\Pi }\right)$ involutions by definition.

The remainder of this section is mostly devoted to proving the following result:

One has $\mathrm{|}Z\mathrm{|}\mathrm{=}{\mathrm{2}}^{c\mathit{}\mathrm{\left(}\mathrm{\Pi }\mathrm{\right)}}$.

We start the proof of this theorem by revisiting Remark 5.

Let V be an $\mathrm{R}$-vector space and let ${X}_{i}\mathrm{,}{X}_{j}\mathrm{\in }\mathrm{End}\mathit{}\mathrm{\left(}V\mathrm{\right)}$ be such that

${X}_{i}^{2}=-{id}_{V}={X}_{j}^{2},{X}_{i}{X}_{j}=-{X}_{j}{X}_{i}.$

Then the map

$\psi :\mathrm{Spin}\left(3\right)\to \mathrm{GL}\left(V\right),$$a+b{e}_{1}{e}_{2}+c{e}_{2}{e}_{3}+d{e}_{1}{e}_{3}↦a{id}_{V}+b{X}_{i}+c{X}_{j}+d{X}_{i}{X}_{j}$

is a group monomorphism such that

$\begin{array}{cc}\hfill \psi \left({\stackrel{~}{\epsilon }}_{12}\left(S\left(\alpha \right)\right)\right)& =\mathrm{cos}\left(\alpha \right){id}_{V}+\mathrm{sin}\left(\alpha \right){X}_{i},\hfill \\ \hfill \psi \left({\stackrel{~}{\epsilon }}_{23}\left(S\left(\alpha \right)\right)\right)& =\mathrm{cos}\left(\alpha \right){id}_{V}+\mathrm{sin}\left(\alpha \right){X}_{j}.\hfill \end{array}$

Proof.

The subspace $ℍ:={〈{id}_{V},{X}_{i},{X}_{j},{X}_{i}{X}_{j}〉}_{ℝ}$ is an $ℝ$-subalgebra of $End\left(V\right)$, the set $\left\{{id}_{V},{X}_{i},{X}_{j},{X}_{i}{X}_{j}\right\}$ is an $ℝ$-basis of $ℍ$, and the $ℝ$-linear extension

$\stackrel{^}{\psi }:\mathrm{Cl}{\left(3\right)}^{0}\to ℍ$

of

$1↦{id}_{V},{e}_{1}{e}_{2}↦{X}_{i},{e}_{1}{e}_{3}↦{X}_{i}{X}_{j},{e}_{2}{e}_{3}↦{X}_{j}$

is an isomorphism of algebras: Indeed, since ${id}_{V}$, ${X}_{i}$, ${X}_{j}$ and ${X}_{i}{X}_{j}$ satisfy the same relations as 1, ${e}_{1}{e}_{2}$, ${e}_{2}{e}_{3}$ and ${e}_{1}{e}_{3}$, the map $\stackrel{^}{\psi }$ is a homomorphism of rings. Since ${X}_{i}\ne {0}_{End\left(V\right)}$, one has $\mathrm{ker}\left(\stackrel{^}{\psi }\right)\ne \mathrm{Cl}{\left(3\right)}^{0}$. By Remark 5, $\mathrm{Cl}{\left(3\right)}^{0}$ is a skew field and, thus, simple as a ring. Therefore, $\stackrel{^}{\psi }$ is injective and, hence, bijective, because ${dim}_{ℝ}ℍ\le 4$, i.e., $\stackrel{^}{\psi }$ is an isomorphism of algebras.

Consequently, the restriction ψ of $\stackrel{^}{\psi }$ to $\mathrm{Spin}\left(3\right)$ is injective with values in the group $\mathrm{GL}\left(V\right)$, i.e., $\psi :\mathrm{Spin}\left(3\right)\to \mathrm{GL}\left(V\right)$ is a group monomorphism. The final statement is immediate from the definitions. ∎

Let V be an $\mathrm{R}$-vector space and let ${X}_{i}\mathrm{,}{X}_{j}\mathrm{\in }\mathrm{End}\mathit{}\mathrm{\left(}V\mathrm{\right)}$ be such that

${X}_{j}\notin {〈{id}_{V},{X}_{i}〉}_{ℝ},{X}_{i}^{2}=-{id}_{V}={X}_{j}^{2},{X}_{i}{X}_{j}={X}_{j}{X}_{i}.$

Then the map

$\psi :〈{\stackrel{~}{\epsilon }}_{12}\left(\mathrm{Spin}\left(2\right)\right),{\stackrel{~}{\epsilon }}_{34}\left(\mathrm{Spin}\left(2\right)\right)〉\subseteq \mathrm{Spin}\left(4\right)\to \mathrm{GL}\left(V\right),$$a+b{e}_{1}{e}_{2}+c{e}_{3}{e}_{4}+d{e}_{1}{e}_{2}{e}_{3}{e}_{4}↦a{id}_{V}+b{X}_{i}+c{X}_{j}+d{X}_{i}{X}_{j}$

is a group monomorphism such that

$\begin{array}{cc}\hfill \psi \left({\stackrel{~}{\epsilon }}_{12}\left(S\left(\alpha \right)\right)\right)& =\mathrm{cos}\left(\alpha \right){id}_{V}+\mathrm{sin}\left(\alpha \right){X}_{i},\hfill \\ \hfill \psi \left({\stackrel{~}{\epsilon }}_{34}\left(S\left(\alpha \right)\right)\right)& =\mathrm{cos}\left(\alpha \right){id}_{V}+\mathrm{sin}\left(\alpha \right){X}_{j}.\hfill \end{array}$

Proof.

The subspace $𝔸:={〈{id}_{V},{X}_{i},{X}_{j},{X}_{i}{X}_{j}〉}_{ℝ}$ is an $ℝ$-subalgebra of $End\left(V\right)$, the set $\left\{{id}_{V},{X}_{i},{X}_{j},{X}_{i}{X}_{j}\right\}$ is an $ℝ$-basis of $𝔸$, and the $ℝ$-linear extension

$\stackrel{^}{\psi }:\stackrel{~}{𝔸}:={〈1,{e}_{1}{e}_{2},{e}_{3}{e}_{4},{e}_{1}{e}_{2}{e}_{3}{e}_{4}〉}_{ℝ}\subseteq \mathrm{Cl}{\left(4\right)}^{0}\to 𝔸$

of

$1↦{id}_{V},{e}_{1}{e}_{2}↦{X}_{i},{e}_{3}{e}_{4}↦{X}_{j},{e}_{1}{e}_{2}{e}_{3}{e}_{4}↦{X}_{i}{X}_{j}$

is an isomorphism of algebras: Indeed, since ${id}_{V}$, ${X}_{i}$, ${X}_{j}$ and ${X}_{i}{X}_{j}$ satisfy the same relations as 1, ${e}_{1}{e}_{2}$, ${e}_{3}{e}_{4}$ and ${e}_{1}{e}_{2}{e}_{3}{e}_{4}$, the map ψ is a homomorphism of rings. The hypothesis ${X}_{j}\notin {〈{id}_{V},{X}_{i}〉}_{ℝ}$ implies that the set $\left\{1,{X}_{i},{X}_{j},{X}_{i}{X}_{j}\right\}$ is $ℝ$-linearly independent. It follows that $\stackrel{^}{\psi }$ is injective and, thus, bijective, because ${dim}_{ℝ}𝔸\le 4$.

Consequently, the restriction ψ of $\stackrel{^}{\psi }$ to $〈{\stackrel{~}{\epsilon }}_{12}\left(\mathrm{Spin}\left(2\right)\right),{\stackrel{~}{\epsilon }}_{34}\left(\mathrm{Spin}\left(2\right)\right)〉\subseteq \mathrm{Spin}\left(4\right)$ is injective with values in $\mathrm{GL}\left(V\right)$, i.e.,

$\psi :〈{\stackrel{~}{\epsilon }}_{12}\left(\mathrm{Spin}\left(2\right)\right),{\stackrel{~}{\epsilon }}_{34}\left(\mathrm{Spin}\left(2\right)\right)〉\subseteq \mathrm{Spin}\left(4\right)\to \mathrm{GL}\left(V\right)$

is a group monomorphism as claimed. The final statement is immediate from the definitions. ∎

We are now in a position to use the results of [20] in order to confirm the conjecture concerning $\mathrm{Spin}\left(\mathrm{\Pi }\right)$ made in [9, footnote 18, p. 24]. The definition of a generalised spin representation can be found in [20, Definition 3.6], the definition and existence of a maximal one in [20, Corollary 3.10].

We point out that [20, Example 3.2] uses a convention for Clifford algebras different from the one used in the present article; however, [20, Corollary 3.10] is formulated and proved without making any reference to Clifford algebras whatsoever.

Let

• Π be an irreducible simply-laced diagram with labelling $\sigma :I\to V$,

• $𝔤$ be the Kac–Moody algebra corresponding to Π and $𝔨$ its maximal compact subalgebra with Berman generators ${Y}_{1}$ , …, ${Y}_{n}$ (cf. [ 20 , Section 1.2] ),

• $\mu :𝔨\to End\left({ℂ}^{s}\right)$, $s\in ℕ$ , be a maximal generalised spin representation (cf. [ 20 , Corollary 3.10] ),

• ${X}_{i}:=2\mu \left({Y}_{i}\right)$ for each $i\in I$.

Then, for each $i\mathrm{\ne }j\mathrm{\in }I$, there exist subgroups ${X}_{i\mathit{}j}\mathrm{\le }{\mathrm{GL}}_{\mathrm{s}}\mathit{}\mathrm{\left(}\mathrm{C}\mathrm{\right)}$ and an enveloping morphism

${\mathrm{\Psi }}_{\mathcal{𝒜}}=\left\{{\psi }_{ij}\mid i\ne j\in I\right\}:\mathcal{𝒜}\left(\mathrm{\Pi },\mathrm{Spin}\left(2\right)\right)\to X:=〈{X}_{ij}\mid i\ne j\in I〉$

with injective ${\psi }_{i\mathit{}j}$ whenever ${\mathrm{\left\{}i\mathrm{,}j\mathrm{\right\}}}^{\sigma }\mathrm{\in }E\mathit{}\mathrm{\left(}\mathrm{\Pi }\mathrm{\right)}$.

Proof.

According to [20, Remark 3.7], given $i\ne j\in I$, one has

${X}_{i}^{2}=-{id}_{V}={X}_{j}^{2},$

and

${X}_{i}{X}_{j}=\left\{\begin{array}{cc}-{X}_{j}{X}_{i},\hfill & \text{if}{\left\{i,j\right\}}^{\sigma }\in E\left(\mathrm{\Pi }\right),\hfill \\ {X}_{j}{X}_{i},\hfill & \text{if}{\left\{i,j\right\}}^{\sigma }\notin E\left(\mathrm{\Pi }\right).\hfill \end{array}$

Moreover, ${X}_{j}\notin {〈{id}_{V},{X}_{i}〉}_{ℝ}$, as μ is maximal. Thus Lemma 11 provides group monomorphisms

${\psi }_{ij}:{\stackrel{~}{K}}_{ij}\to {\mathrm{GL}}_{s}\left(ℂ\right),$

if ${\left\{i,j\right\}}^{\sigma }\in E\left(\mathrm{\Pi }\right)$, and Lemma 12 provides group homomorphisms

${\psi }_{ij}:{\stackrel{~}{K}}_{ij}\to {\mathrm{GL}}_{s}\left(ℂ\right)$

with kernel $〈\left(-1,-1\right)〉$, if ${\left\{i,j\right\}}^{\sigma }\notin E\left(\mathrm{\Pi }\right)$. This allows one to define

${X}_{ij}:=\mathrm{im}\left({\psi }_{ij}\right).$

Restriction of the ranges of the maps ${\psi }_{ij}$ to ${X}_{ij}$ thus provides group isomorphisms

${\psi }_{ij}:{\stackrel{~}{K}}_{ij}=\mathrm{Spin}\left(3\right)\to {X}_{ij},$

if ${\left\{i,j\right\}}^{\sigma }\in E\left(\mathrm{\Pi }\right)$, and group epimorphisms

${\psi }_{ij}:{\stackrel{~}{K}}_{ij}=\mathrm{Spin}\left(2\right)×\mathrm{Spin}\left(2\right)\to {X}_{ij},$

if ${\left\{i,j\right\}}^{\sigma }\notin E\left(\mathrm{\Pi }\right)$, satisfying for all $i\ne j\in I$,

${\psi }_{ij}\left({\stackrel{~}{\varphi }}_{ij}^{j}\left(\mathrm{cos}\left(\alpha \right)+\mathrm{sin}\left(\alpha \right){e}_{1}{e}_{2}\right)\right)=\mathrm{cos}\left(\alpha \right){id}_{V}+\mathrm{sin}\left(\alpha \right){X}_{j}.$

In particular, one has for all $i\ne j\ne k$,

${\psi }_{ij}\circ {\stackrel{~}{\varphi }}_{ij}^{j}={\psi }_{kj}\circ {\stackrel{~}{\varphi }}_{kj}^{j}.$

The set ${\mathrm{\Psi }}_{\mathcal{𝒜}}:=\left\{{\psi }_{ij}\mid i\ne j\in I\right\}$ is the desired enveloping morphism. ∎

Let everything be as in Theorem 14. By universality of

${\tau }_{\mathrm{Spin}\left(\mathrm{\Pi }\right)}:\mathcal{𝒜}\left(\mathrm{\Pi },\mathrm{Spin}\left(2\right)\right)\to \mathrm{Spin}\left(\mathrm{\Pi }\right)$

(cf. Definition 5) there exists an epimorphism

$\mathrm{\Xi }:\mathrm{Spin}\left(\mathrm{\Pi }\right)\to X$

such that the following diagram commutes:

The commutative diagram in Lemma 6 and the finiteness of the central extension $\mathrm{Spin}\left(\mathrm{\Pi }\right)\to K\left(\mathrm{\Pi }\right)$ by Observation 9 in fact allow one to lift the topological universality statement from Remark 4 concerning

${\tau }_{K\left(\mathrm{\Pi }\right)}:\mathcal{𝒜}\left(\mathrm{\Pi },\mathrm{SO}\left(2\right)\right)\to K\left(\mathrm{\Pi }\right)$

to a topological universality statement concerning

${\tau }_{\mathrm{Spin}\left(\mathrm{\Pi }\right)}:\mathcal{𝒜}\left(\mathrm{\Pi },\mathrm{Spin}\left(2\right)\right)\to \mathrm{Spin}\left(\mathrm{\Pi }\right).$

Moreover, the maps ψ constructed in Lemmas 11 and 12 are certainly continuous with respect to the Lie group topologies, if ${dim}_{ℝ}\left(V\right)<\mathrm{\infty }$.

In particular, the enveloping morphism ${\mathrm{\Psi }}_{\mathcal{𝒜}}=\left\{{\psi }_{ij}\right\}$ from the theorem consists of continuous maps, so that by universality $\mathrm{\Xi }:\mathrm{Spin}\left(\mathrm{\Pi }\right)\to X$ is continuous as well.

As an immediate consequence we record:

Let Π be an irreducible simply-laced diagram. Then one has ${\mathrm{1}}_{\mathrm{Spin}\mathit{}\mathrm{\left(}\mathrm{\Pi }\mathrm{\right)}}\mathrm{\ne }\mathrm{-}{\mathrm{1}}_{\mathrm{Spin}\mathit{}\mathrm{\left(}\mathrm{\Pi }\mathrm{\right)}}$.

Proof.

We conclude from Remark 15

$\begin{array}{cc}\hfill \mathrm{\Xi }\left(-{1}_{\mathrm{Spin}\left(\mathrm{\Pi }\right)}\right)& =\left(\mathrm{\Xi }\circ {\tau }_{12}\circ {\stackrel{~}{\varphi }}_{12}^{1}\right)\left(-{1}_{\mathrm{Spin}\left(2\right)}\right)=\left({\psi }_{12}\circ {\stackrel{~}{\varphi }}_{12}^{1}\right)\left(-{1}_{\mathrm{Spin}\left(2\right)}\right)\hfill \\ & =\mathrm{cos}\left(\pi \right){id}_{V}+\mathrm{sin}\left(\pi \right){X}_{1}=-{id}_{V},\hfill \end{array}$

and, hence, ${1}_{\mathrm{Spin}\left(\mathrm{\Pi }\right)}\ne -{1}_{\mathrm{Spin}\left(\mathrm{\Pi }\right)}$. ∎

Let Π be a simply-laced diagram. Then the universal enveloping group $\mathrm{\left(}\mathrm{Spin}\mathit{}\mathrm{\left(}\mathrm{\Pi }\mathrm{\right)}\mathrm{,}{\tau }_{\mathrm{Spin}\mathit{}\mathrm{\left(}\mathrm{\Pi }\mathrm{\right)}}\mathrm{=}\mathrm{\left\{}{\tau }_{i\mathit{}j}\mathrm{\mid }i\mathrm{\ne }j\mathrm{\in }I\mathrm{\right\}}\mathrm{\right)}$ of $\mathcal{A}\mathit{}\mathrm{\left(}\mathrm{\Pi }\mathrm{,}\mathrm{Spin}\mathit{}\mathrm{\left(}\mathrm{2}\mathrm{\right)}\mathrm{\right)}$ is a ${\mathrm{2}}^{c\mathit{}\mathrm{\left(}\mathrm{\Pi }\mathrm{\right)}}$-fold central extension of the universal enveloping group $K\mathit{}\mathrm{\left(}\mathrm{\Pi }\mathrm{\right)}$ of $\mathcal{A}\mathit{}\mathrm{\left(}\mathrm{\Pi }\mathrm{,}\mathrm{SO}\mathit{}\mathrm{\left(}\mathrm{2}\mathrm{\right)}\mathrm{\right)}$.

Proof.

Let ${\mathrm{\Pi }}_{1},\mathrm{\dots },{\mathrm{\Pi }}_{c\left(\mathrm{\Pi }\right)}$ be the connected components of Π. Then

$\mathrm{Spin}\left(\mathrm{\Pi }\right)=\mathrm{Spin}\left({\mathrm{\Pi }}_{1}\right)×\mathrm{\cdots }×\mathrm{Spin}\left({\mathrm{\Pi }}_{c\left(\mathrm{\Pi }\right)}\right).$

Indeed,

${\tau }_{\mathrm{Spin}\left(\mathrm{\Pi }\right)}:\mathcal{𝒜}\left(\mathrm{\Pi },\mathrm{Spin}\left(2\right)\right)\to \mathrm{Spin}\left({\mathrm{\Pi }}_{1}\right)×\mathrm{\cdots }×\mathrm{Spin}\left({\mathrm{\Pi }}_{c\left(\mathrm{\Pi }\right)}\right)$

with

${\tau }_{ij}:{\stackrel{~}{K}}_{ij}\to {\tau }_{ij}\left({\stackrel{~}{K}}_{ij}\right)$

if $\mathcal{𝒦}\left(i\right)=\mathcal{𝒦}\left(j\right)$ and

${\tau }_{ij}:{\stackrel{~}{K}}_{ij}\to \left({\tau }_{ij}\circ {\stackrel{~}{\varphi }}_{ij}^{i}\right)\left(\mathrm{Spin}\left(2\right)\right)×\left({\tau }_{ij}\circ {\stackrel{~}{\varphi }}_{ij}^{j}\right)\left(\mathrm{Spin}\left(2\right)\right),\left(x,y\right)↦{\tau }_{ij}\left(x,y\right)$

if $\mathcal{𝒦}\left(i\right)\ne \mathcal{𝒦}\left(j\right)$ is an enveloping morphism.

It therefore suffices to prove the theorem for irreducible simply-laced diagrams Π. In this case, however, it is immediate from Proposition 9 applied to the $\stackrel{~}{U}$-amalgam $\mathcal{𝒜}\left(\mathrm{\Pi },\mathrm{Spin}\left(2\right)\right)$ and the U-amalgam $\mathcal{𝒜}\left(\mathrm{\Pi },\mathrm{SO}\left(2\right)\right)$ with $\stackrel{~}{U}=\mathrm{Spin}\left(2\right)$, $U=\mathrm{SO}\left(2\right)$ and $\stackrel{~}{V}=〈-1〉$ combined with Lemma 7 and Corollary 16. ∎

We have proved Theorem 10 and Theorem A from the introduction.

12 Strategies for reducing the general case to the simply-laced one

Until now we exclusively studied spin covers of maximal compact subgroups of split real Kac–Moody groups of simply-laced type. Our next goal is to generalise this concept to arbitrary Dynkin diagrams resp. generalised Cartan matrices. We pursue this goal via two strategies: The first one is via epimorphisms between maximal compact subgroups induced by local epimorphisms on amalgam-level in rank two where we replace non-simple edges by non-edges, simple edges or double edges; the second one is via embeddings into larger groups by unfolding the diagrams resp. the Cartan matrices to simply-laced cover diagrams as in [20].

The first strategy will allow us to transform arbitrary Dynkin diagrams resp. generalised Cartan matrices into doubly-laced ones. The second strategy will work for the resulting doubly-laced generalised Cartan matrices. A combination of both strategies allows us to deal with arbitrary generalised Cartan matrices.

In order to deal with the two non-simply-laced spherical diagrams of rank two, ${\mathrm{C}}_{2}$ and ${\mathrm{G}}_{2}$, we consider point-line models of the Tits buildings of the split real Lie groups ${\mathrm{Sp}}_{4}\left(ℝ\right)$ and ${\mathrm{G}}_{2}\left(2\right)$, the so-called symplectic quadrangle and the so-called split Cayley hexagon. As in the proof of Theorem 9, the Iwasawa decomposition implies that the maximal compact subgroups ${\mathrm{U}}_{2}\left(ℂ\right)\le {\mathrm{Sp}}_{4}\left(ℝ\right)$ and $\mathrm{SO}\left(4\right)\le {\mathrm{G}}_{2}\left(2\right)$ act flag-transitively on the respective point-line geometries.

Their unique double covers $\mathrm{SO}\left(2\right)×{\mathrm{SU}}_{2}\left(ℂ\right)↠{\mathrm{U}}_{2}\left(ℂ\right)$ and $\mathrm{Spin}\left(4\right)↠\mathrm{SO}\left(4\right)$ fit into the commutative diagrams

and

which allow one to transform point and line stabilisers in ${\mathrm{U}}_{2}\left(ℂ\right)$ and $\mathrm{SO}\left(4\right)$ into point, resp. line stabilisers in $\mathrm{SO}\left(3\right)$ in a way that is compatible with the covering maps. This in turn will allow us to transform a $\mathrm{Spin}\left(2\right)$-amalgam for a given two-spherical diagram Π into a $\mathrm{Spin}\left(2\right)$-amalgam for the simply-laced diagram ${\mathrm{\Pi }}^{\mathrm{sl}}$ that one obtains from Π by replacing all edges by simple edges. As a consequence – based on Theorem 17 – in Theorem 1 below we will be able to prove that the spin cover $\mathrm{Spin}\left(\mathrm{\Pi }\right)$ is a non-trivial central extension of $K\left(\mathrm{\Pi }\right)$ for suitable two-spherical diagrams Π.

As a caveat we point out that the compatibility of the covering maps in the ${\mathrm{C}}_{2}$ case is quite subtle and actually fails under certain circumstances, due to the phenomena described in Lemma 10. In order to control these subtleties we introduce the notion of admissible colourings of Dynkin diagrams in Definition 2. These subtleties are also why we actually only replace certain double edges by single edges and additionally employ Strategy 2 below.

When trying to deal with non-two-spherical diagrams further subtleties arise. The non-spherical Cartan matrices of rank two are of the form

$\left(\begin{array}{cc}\hfill 2\hfill & \hfill -r\hfill \\ \hfill -s\hfill & \hfill 2\hfill \end{array}\right)$

for $r,s\in ℕ$ such that $rs\ge 4$. The isomorphism type of the maximal compact subgroup K of the corresponding split real Kac–Moody group depends (only) on the parities of r and s. Indeed, in all cases K is isomorphic to a free amalgamated product

$K\cong {K}_{1}{T}_{K}{*}_{{T}_{K}}{K}_{2}{T}_{K},$

where

${K}_{1}\cong \mathrm{SO}\left(2\right)\cong {K}_{2}$

with ${T}_{K}=\left\{1,{t}_{1},{t}_{2},{t}_{1}{t}_{2}\right\}\cong ℤ/2ℤ×ℤ/2ℤ$ and ${T}_{K}\cap {K}_{1}=〈{t}_{1}〉$, ${T}_{K}\cap {K}_{2}=〈{t}_{2}〉$ and ${K}_{i}⊴{K}_{i}{T}_{K}$. We conclude that the isomorphism type of K is known once the action of ${t}_{1}$ on ${K}_{2}$ and the action of ${t}_{2}$ on ${K}_{1}$ are known. It turns out that ${t}_{1}$ centralises ${K}_{2}$ if and only if r is even and inverts ${K}_{2}$ if and only if r is odd; similarly, ${t}_{2}$ centralises ${K}_{1}$ if and only if s is even and inverts ${K}_{1}$ if and only if s is odd (cf. Remark 4).

To these four cases of parities of r and s correspond three cases of epimorphisms from K onto compact Lie groups: $K↠\mathrm{SO}\left(2\right)×\mathrm{SO}\left(2\right)$, if both r and s are even; $K↠\mathrm{SO}\left(3\right)$, if both r and s are odd; $K↠{\mathrm{U}}_{2}\left(ℂ\right)$, if r and s have different parities. A study of various double covers of K will, in analogy to what we sketched above for diagrams of type ${\mathrm{C}}_{2}$ and ${\mathrm{G}}_{2}$, enable us to replace edges labelled $\mathrm{\infty }$ by non-edges, simple edges, resp. double edges, thus allowing us to understand the non-two-spherical situation as well. Again, the case in which r and s have different parities lead to some subtleties that we get control of with the concept of admissible colourings introduced in Definition 2.

Following this strategy leads directly to Proposition 11.

Let Π with type set I and (generalised) Cartan matrix

$A={\left(a\left(i,j\right)\right)}_{i,j\in I}$

be an irreducible doubly-laced diagram that admits two root lengths. Then the unfolded Dynkin diagram is the simply-laced Dynkin diagram ${\mathrm{\Pi }}^{\mathrm{un}}$ with type set

${I}^{\mathrm{un}}:=\left\{±i\mid i\in I,i\text{short root}\right\}\cup \left\{i\mid i\in I,i\text{long root}\right\}$

and edges defined via the generalised Cartan matrix ${A}^{\mathrm{un}}={\left({a}^{\mathrm{un}}\left(i,j\right)\right)}_{i,j\in {I}^{\mathrm{un}}}$ given by

${a}^{\mathrm{un}}\left(i,j\right)=\left\{\begin{array}{cc}0,\hfill & \text{if}|i|\text{,}|j|\text{have different lengths and}a\left(|i|,|j|\right)=0,\hfill \\ -1,\hfill & \text{if}|i|\text{,}|j|\text{have different lengths and}a\left(|i|,|j|\right)\ne 0,\hfill \\ a\left(|i|,|j|\right),\hfill & \text{if}|i|\text{,}|j|\text{have the same length and}ij>0,\hfill \\ 0,\hfill & \text{if}|i|\text{,}|j|\text{have the same length and}ij<0,\hfill \end{array}$

(cf. Definition 2).

There exists an embedding of $K\left(\mathrm{\Pi }\right)$ into $K\left({\mathrm{\Pi }}^{\mathrm{un}}\right)$ that by Corollary 6 allows one to related the respective spin covers to one another.

13 Diagrams of type ${\mathrm{G}}_{2}$

In this section we prepare Strategy 1 for diagrams of type ${\mathrm{G}}_{2}$.

Denote by $ℍ:=\left\{a+bi+cj+dk\mid a,b,c,d\in ℝ\right\}$ the real quaternions. Then the standard involution of $ℍ$ is given by

$\overline{\cdot }:ℍ\to ℍ,x=a+bi+cj+dk↦\overline{x}=a-bi-cj-dk.$

The set of purely imaginary quaternions, cf. [37, 11.6], is

$Puℍ:=\left\{x\in ℍ\mid x=-\overline{x}\right\}=\left\{bi+cj+dk\mid b,c,d\in ℝ\right\}\subset ℍ.$

The split Cayley algebra $𝕆$ is defined as the vector space $ℍ\oplus ℍ$ endowed with the multiplication

$xy=\left({x}_{1},{x}_{2}\right)\left({y}_{1},{y}_{2}\right)=\left({x}_{1}{y}_{1}+{y}_{2}\overline{{x}_{2}},{y}_{1}{x}_{2}+\overline{{x}_{1}}{y}_{2}\right),$

cf. [7, Section 5.1]. The real split Cayley hexagon $\mathcal{ℋ}\left(ℝ\right)$ consists of the one- and two-dimensional real subspaces of $𝕆$ for which the restriction of the multiplication map is trivial, i.e., $\mathcal{ℋ}\left(ℝ\right)=\left(\mathcal{𝒫},\mathcal{ℒ},\subset \right)$ with the point set

$\mathcal{𝒫}:=\left\{{〈x〉}_{ℝ}\mid x\in 𝕆,{x}^{2}=0\ne x\right\}$

and the line set

$\mathcal{ℒ}:=\left\{{〈x,y〉}_{ℝ}\mid {〈x〉}_{ℝ}\ne {〈y〉}_{ℝ}\in \mathcal{𝒫},xy=0\right\},$

cf. [7, Section 5.1], also [44, Section 2.4.9].

Let $x\mathrm{=}\mathrm{\left(}{x}_{\mathrm{1}}\mathrm{,}{x}_{\mathrm{2}}\mathrm{\right)}\mathrm{\in }\mathrm{O}$. Then ${x}^{\mathrm{2}}\mathrm{=}\mathrm{0}$ if and only if ${x}_{\mathrm{1}}\mathrm{\in }\mathrm{Pu}\mathit{}\mathrm{H}$ and $\overline{{x}_{\mathrm{1}}}\mathit{}{x}_{\mathrm{1}}\mathrm{-}\overline{{x}_{\mathrm{2}}}\mathit{}{x}_{\mathrm{2}}\mathrm{=}\mathrm{0}$.

Proof.

Suppose ${x}_{1}\in Puℍ$ and $\overline{{x}_{1}}{x}_{1}-\overline{{x}_{2}}{x}_{2}=0$. Then

$-{x}_{1}{x}_{1}-{x}_{2}\overline{{x}_{2}}=\overline{{x}_{1}}{x}_{1}-\overline{{x}_{2}}{x}_{2}=0$

and

${x}_{1}{x}_{2}+\overline{{x}_{1}}{x}_{2}={x}_{1}{x}_{2}-{x}_{1}{x}_{2}=0,$

and, thus, ${x}^{2}=0$. Conversely, suppose ${x}^{2}=0$. Then

$0={x}^{2}=\left({x}_{1}{x}_{1}+{x}_{2}\overline{{x}_{2}},{x}_{1}{x}_{2}+\overline{{x}_{1}}{x}_{2}\right),$

so if ${x}_{2}=0$, then ${x}_{1}=0$, and there is nothing to show. For ${x}_{2}\ne 0$ multiplication of ${x}_{1}{x}_{2}+\overline{{x}_{1}}{x}_{2}=0$ from the right with ${x}_{2}^{-1}$ gives ${x}_{1}+\overline{{x}_{1}}=0$ or, equivalently, ${x}_{1}\in Puℍ$. But now $0={x}_{1}{x}_{1}+{x}_{2}\overline{{x}_{2}}=-\overline{{x}_{1}}{x}_{1}+\overline{{x}_{2}}{x}_{2}$, and the claim follows. ∎

Let $N:ℍ\to ℝ$, $x↦x\overline{x}$ be the norm map associated to the standard involution of the real quaternions. Moreover, let

${\mathrm{U}}_{1}\left(ℍ\right):=\left\{x\in ℍ\mid x\overline{x}=1\right\}$

be the group of real quaternions of norm one.

By Remark 3 (see also [37, Lemma 11.22]), the group $\mathrm{SO}\left(4\right)$ is isomorphic to the group consisting of the maps

$ℍ\to ℍ,x↦ax{b}^{-1}\mathit{ }\text{for}a,b\in {\mathrm{U}}_{1}\left(ℍ\right).$

The group $\mathrm{SO}\mathit{}\mathrm{\left(}\mathrm{4}\mathrm{\right)}\mathrm{\cong }\mathrm{\left\{}\mathrm{H}\mathrm{\to }\mathrm{H}\mathrm{,}x\mathrm{↦}a\mathit{}x\mathit{}{b}^{\mathrm{-}\mathrm{1}}\mathrm{\mid }a\mathrm{,}b\mathrm{\in }{\mathrm{U}}_{\mathrm{1}}\mathit{}\mathrm{\left(}\mathrm{H}\mathrm{\right)}\mathrm{\right\}}$ acts flag-transitively on the split Cayley hexagon $\mathcal{H}\mathit{}\mathrm{\left(}\mathrm{R}\mathrm{\right)}$ via

${〈\left({x}_{1},{x}_{2}\right)〉}_{ℝ}↦{〈\left(a{x}_{1}{a}^{-1},a{x}_{2}{b}^{-1}\right)〉}_{ℝ}.$

Proof.

We show that the map

${f}_{a,b}:𝕆\to 𝕆:\left({x}_{1},{x}_{2}\right)↦\left(a{x}_{1}{a}^{-1},a{x}_{2}{b}^{-1}\right)$

is an algebra automorphism of $𝕆$ for all $a,b\in {\mathrm{U}}_{1}\left(ℍ\right)$. Then it also induces an automorphism of $\mathcal{ℋ}\left(ℝ\right)$, because it is defined via the multiplication in $𝕆$. We have

${f}_{a,b}\left({x}_{1},{x}_{2}\right)\cdot {f}_{a,b}\left({y}_{1},{y}_{2}\right)=\left(a{x}_{1}{a}^{-1},a{x}_{2}{b}^{-1}\right)\cdot \left(a{y}_{1}{a}^{-1},a{y}_{2}{b}^{-1}\right)$$=\left(\left(a{x}_{1}{a}^{-1}\right)\left(a{y}_{1}{a}^{-1}\right)+\left(a{y}_{2}{b}^{-1}\right)\overline{\left(a{x}_{2}{b}^{-1}\right)},$$\left(a{y}_{1}{a}^{-1}\right)\left(a{x}_{2}{b}^{-1}\right)+\overline{\left(a{x}_{1}{a}^{-1}\right)}\left(a{y}_{2}{b}^{-1}\right)\right)$$=\left(a\left({x}_{1}{y}_{1}+{y}_{2}\overline{{x}_{2}}\right){a}^{-1},a\left({y}_{1}{x}_{2}+\overline{{x}_{1}}{y}_{2}\right){b}^{-1}\right)$$={f}_{a,b}\left(\left({x}_{1},{x}_{2}\right)\left({y}_{1},{y}_{2}\right)\right),$

whence the map ${f}_{a,b}$ is multiplicative. Since it is certainly an $ℝ$-linear bijection, it is an algebra automorphism of $𝕆$. Flag-transitivity is an immediate consequence of the Iwasawa decomposition and the fact that $\mathrm{SO}\left(4\right)$ is the maximal compact subgroup of the (simply connected semisimple) split real group ${\mathrm{G}}_{2}\left(2\right)$ of type ${\mathrm{G}}_{2}$. ∎

There exists a nice direct proof of flag-transitivity without making use of the Iwasawa decomposition and the structure theory of ${\mathrm{G}}_{2}\left(2\right)$ that in particular illustrates how to compute point and line stabilisers and, thus, helps our understanding of how to properly embed the circle group into $\mathrm{SO}\left(4\right)$ for our amalgamation problem.

Let

$Pu𝕆:=Puℍ\oplus ℍ$

be the set of purely imaginary split octonions and consider the points of the real (projective) quadric

$\overline{{x}_{1}}{x}_{1}-\overline{{x}_{2}}{x}_{2}=0$

in $Pu𝕆$, i.e., the set of isotropic one-dimensional real subspaces of $Pu𝕆$. By Remark 3 (see also [37, 11.24]), the group $\mathrm{SO}\left(3\right)$ is isomorphic to the group consisting of the maps

$Puℍ\to Puℍ:x↦ax{a}^{-1}\text{for}a\in {\mathrm{U}}_{1}\left(ℍ\right)$

and acts transitively on the set $\left\{\left\{x,-x\right\}\subset Puℍ\mid x\overline{x}=1\right\}$. Moreover, for each $a,x,z\in {\mathrm{U}}_{1}\left(ℍ\right)$, there exists a unique solution $b\in {\mathrm{U}}_{1}\left(ℍ\right)$ for the equation

$z=ax{b}^{-1}.$

Hence $\mathrm{SO}\left(4\right)$ acts transitively on the set

$\left\{\left\{\left(x,y\right),\left(-x,-y\right)\right\}\subset Puℍ×ℍ\mid x\overline{x}=1=y\overline{y}\right\}.$

But this implies point transitivity on the projective real quadric $\overline{{x}_{1}}{x}_{1}-\overline{{x}_{2}}{x}_{2}=0$ in $Pu𝕆$, which, in turn, implies point transitivity on $\mathcal{ℋ}\left(ℝ\right)$ by Lemma 3.

Now choose one point of $\mathcal{ℋ}\left(ℝ\right)$, say ${〈\left(i,i\right)〉}_{ℝ}$. Then a point ${〈y〉}_{ℝ}={〈\left({y}_{1},{y}_{2}\right)〉}_{ℝ}$ is collinear to this point if and only if

$\left(i{y}_{1}-{y}_{2}i,{y}_{1}i-i{y}_{2}\right)=0⇔{y}_{1}=-i{y}_{2}i.$

So the question of transitivity of the stabiliser of ${〈\left(i,i\right)〉}_{ℝ}$ in $\mathrm{SO}\left(4\right)$ on the line pencil of ${〈\left(i,i\right)〉}_{ℝ}$ in $\mathcal{ℋ}\left(ℝ\right)$ is equivalent to the question of transitivity of the stabiliser of ${〈i〉}_{ℝ}$ in $\mathrm{SO}\left(3\right)$ on the line pencil of ${〈i〉}_{ℝ}$ in the projective plane $Puℍ\cong Puℍ\oplus \left\{0\right\}\subset Pu𝕆$.

But since the latter is transitive, so is the former, and hence $\mathrm{SO}\left(4\right)$ acts flag-transitively on $\mathcal{ℋ}\left(ℝ\right)$ by means of the maps given in Lemma 5.

We denote the embeddings of the circle group into $\mathrm{SO}\left(4\right)$ as panel stabilisers (stabilisers of line pencils or point rows) of the split Cayley hexagon by ${\eta }_{p}$ resp. ${\eta }_{l}$. Concretely, one has up to choosing an orientation of the circle group and up to choosing the split Cayley hexagon or its dual,

${\eta }_{p}:\mathrm{SO}\left(2\right)\to \mathrm{SO}\left(4\right),$

given by

$D\left(\alpha \right)↦\left(\begin{array}{cc}\hfill {I}_{2}\hfill & \hfill \hfill \\ \hfill \hfill & \hfill D\left(\alpha \right)\hfill \end{array}\right)={\epsilon }_{34}\left(D\left(\alpha \right)\right),$

and

${\eta }_{l}:\mathrm{SO}\left(2\right)\to \mathrm{SO}\left(4\right),$

given by

$D\left(\alpha \right)↦\stackrel{~}{D}\left(\alpha \right):=\left(\begin{array}{cccc}\hfill \mathrm{cos}\left(2\alpha \right)\hfill & \hfill \hfill & \hfill \hfill & \hfill -\mathrm{sin}\left(2\alpha \right)\hfill \\ \hfill \hfill & \hfill \mathrm{cos}\left(\alpha \right)\hfill & \hfill \mathrm{sin}\left(\alpha \right)\hfill & \hfill \hfill \\ \hfill \hfill & \hfill -\mathrm{sin}\left(\alpha \right)\hfill & \hfill \mathrm{cos}\left(\alpha \right)\hfill & \hfill \hfill \\ \hfill \mathrm{sin}\left(2\alpha \right)\hfill & \hfill \hfill & \hfill \hfill & \hfill \mathrm{cos}\left(2\alpha \right)\hfill \end{array}\right)={\epsilon }_{14}\left(D\left(2\alpha \right)\right)\cdot {\epsilon }_{23}\left(D\left(-\alpha \right)\right).$

Let $B\mathrm{:=}\mathrm{diag}\mathit{}\mathrm{\left(}\mathrm{-}\mathrm{1}\mathrm{,}\mathrm{1}\mathrm{,}\mathrm{1}\mathrm{,}\mathrm{-}\mathrm{1}\mathrm{\right)}$, $C\mathrm{:=}\mathrm{diag}\mathit{}\mathrm{\left(}\mathrm{-}\mathrm{1}\mathrm{,}\mathrm{-}\mathrm{1}\mathrm{,}\mathrm{1}\mathrm{,}\mathrm{1}\mathrm{\right)}\mathrm{\in }{\mathrm{U}}_{\mathrm{2}}\mathit{}\mathrm{\left(}\mathrm{C}\mathrm{\right)}$. Then the following hold:

• (a)

The map ${\gamma }_{B}:\mathrm{SO}\left(4\right)\to \mathrm{SO}\left(4\right)$, $A↦B\cdot A\cdot {B}^{-1}=B\cdot A\cdot B$ is an automorphism of $\mathrm{SO}\left(4\right)$ such that

${\gamma }_{B}\circ {\eta }_{p}={\eta }_{p}\circ \mathrm{inv}\mathit{ }\text{𝑎𝑛𝑑}\mathit{ }{\gamma }_{B}\circ {\eta }_{l}={\eta }_{l}.$

• (b)

The map ${\gamma }_{C}:\mathrm{SO}\left(4\right)\to \mathrm{SO}\left(4\right)$, $A↦C\cdot A\cdot {C}^{-1}=C\cdot A\cdot C$ is an automorphism of $\mathrm{SO}\left(4\right)$ such that

${\gamma }_{C}\circ {\eta }_{p}={\eta }_{p}\mathit{ }\text{𝑎𝑛𝑑}\mathit{ }{\gamma }_{C}\circ {\eta }_{l}={\eta }_{l}\circ \mathrm{inv}.$

Proof.

Straightforward. ∎

In the following, let

${\stackrel{~}{\eta }}_{p}:\mathrm{Spin}\left(2\right)\to \mathrm{Spin}\left(4\right),$$S\left(\alpha \right)↦{\stackrel{~}{\epsilon }}_{34}\left(S\left(\alpha \right)\right),$${\stackrel{~}{\eta }}_{l}:\mathrm{Spin}\left(2\right)\to \mathrm{Spin}\left(4\right),$$S\left(\alpha \right)↦{\stackrel{~}{\epsilon }}_{14}\left(S\left(2\alpha \right)\right)\cdot {\stackrel{~}{\epsilon }}_{23}\left(S\left(-\alpha \right)\right),$

and recall from Theorem 8 (b) that for $n\ge 2$ the map ${\rho }_{n}:\mathrm{Spin}\left(n\right)\to \mathrm{SO}\left(n\right)$ is the twisted adjoint representation.

In order to generalise our definition of spin amalgams, we need ${\stackrel{~}{\eta }}_{p}$ and ${\stackrel{~}{\eta }}_{l}$ to be injective. For the former this is clear from its definition, for the latter we verify it now.

The map ${\stackrel{\mathrm{~}}{\eta }}_{l}$ is a monomorphism.

Proof.

For $S\left(\alpha \right)\in \mathrm{ker}{\stackrel{~}{\eta }}_{l}$ one has

${\stackrel{~}{\eta }}_{l}\left(S\left(\alpha \right)\right)=\left(\mathrm{cos}\left(2\alpha \right)+\mathrm{sin}\left(2\alpha \right){e}_{1}{e}_{4}\right)\left(\mathrm{cos}\left(\alpha \right)-\mathrm{sin}\left(\alpha \right){e}_{2}{e}_{3}\right)=1$

and, thus, $\alpha \in \pi ℤ$. As ${\stackrel{~}{\eta }}_{l}\left(S\left(\pi \right)\right)=-1$ and ${\stackrel{~}{\eta }}_{l}\left(S\left(2\pi \right)\right)=1$, one obtains

$\mathrm{ker}{\stackrel{~}{\eta }}_{l}=\left\{S\left(\alpha \right)\mid \alpha \in 2\pi ℤ\right\}=\left\{1\right\}.\mathit{∎}$

One has

${\rho }_{4}\circ {\stackrel{~}{\eta }}_{p}={\eta }_{p}\circ {\rho }_{2},{\rho }_{4}\circ {\stackrel{~}{\eta }}_{l}={\eta }_{l}\circ {\rho }_{2}.$

Proof.

For $\alpha \in ℝ$,

$\left({\rho }_{4}\circ {\stackrel{~}{\eta }}_{p}\right)\left(S\left(\alpha \right)\right)=\left({\rho }_{4}\circ {\stackrel{~}{\epsilon }}_{34}\right)\left(S\left(\alpha \right)\right)=\left({\epsilon }_{34}\circ \rho \right)\left(S\left(\alpha \right)\right)=\left({\eta }_{p}\circ \rho \right)\left(S\left(\alpha \right)\right)$

and

$\left({\rho }_{4}\circ {\stackrel{~}{\eta }}_{l}\right)\left(S\left(\alpha \right)\right)={\rho }_{4}\left({\stackrel{~}{\epsilon }}_{14}\left(S\left(2\alpha \right)\right)\cdot {\stackrel{~}{\epsilon }}_{23}\left(S\left(-\alpha \right)\right)\right)$$={\epsilon }_{14}\left(D\left(4\alpha \right)\right)\cdot {\epsilon }_{23}\left(D\left(-2\alpha \right)\right)$$=\left({\eta }_{l}\circ {\rho }_{2}\right)\left(S\left(\alpha \right)\right).\mathit{∎}$

Let $V\mathrm{:=}\mathrm{H}$ and $\mathcal{E}\mathrm{:=}\mathrm{\left\{}\mathrm{1}\mathrm{,}i\mathrm{,}j\mathrm{,}k\mathrm{\right\}}$. Then the following hold:

• (a)

For $a,b\in {\mathrm{U}}_{1}\left(ℍ\right)$ the maps

${\mathrm{\ell }}_{a}:ℍ\to ℍ,x↦ax\mathit{ }\text{𝑎𝑛𝑑}\mathit{ }{r}_{b}:ℍ\to ℍ,x↦x{b}^{-1}$

preserve the norm

$N:ℍ\to ℝ,x↦x\overline{x}.$

In particular, ${\mathrm{\ell }}_{a},{r}_{b}\in \mathrm{SO}\left(ℍ\right)\cong \mathrm{SO}\left(4\right)$.

• (b)

The map

${\mathrm{U}}_{1}\left(ℍ\right)×{\mathrm{U}}_{1}\left(ℍ\right)\to \mathrm{SO}\left(4\right),\left(a,b\right)↦{\mathrm{\ell }}_{a}\circ {r}_{b}$

is a group epimorphism with kernel $\left\{\left(1,1\right),\left(-1,-1\right)\right\}$.

Proof.

This has been discussed in Remark 3. Alternatively, it also follows from [37, Lemma 11.22 to Corollary 11.25]. ∎

For $\mathcal{ℰ}=\left\{1,i,j,k\right\}$ define

${L}_{a}:={M}_{\mathcal{ℰ}}\left({\mathrm{\ell }}_{a}\right)\in \mathrm{SO}\left(4\right),{R}_{b}:={M}_{\mathcal{ℰ}}\left({r}_{b}\right)\in \mathrm{SO}\left(4\right).$

We observe the following.

• (a)

The map ${\mathrm{U}}_{1}\left(ℍ\right)×{\mathrm{U}}_{1}\left(ℍ\right)\to \mathrm{SO}\left(4\right)$ from Lemma 7(b) equals the covering map ${\rho }_{4}$, cf. Remark 4.

• (b)

Given $x=a+bi+cj+dk\in {\mathrm{U}}_{1}\left(ℍ\right)$, one has ${x}^{-1}=a-bi-cj-dk$, and a short computation shows

${L}_{x}=\left(\begin{array}{cccc}\hfill a\hfill & \hfill -b\hfill & \hfill -c\hfill & \hfill -d\hfill \\ \hfill b\hfill & \hfill a\hfill & \hfill -d\hfill & \hfill c\hfill \\ \hfill c\hfill & \hfill d\hfill & \hfill a\hfill & \hfill -b\hfill \\ \hfill d\hfill & \hfill -c\hfill & \hfill b\hfill & \hfill a\hfill \end{array}\right),{R}_{x}=\left(\begin{array}{cccc}\hfill a\hfill & \hfill b\hfill & \hfill c\hfill & \hfill d\hfill \\ \hfill -b\hfill & \hfill a\hfill & \hfill -d\hfill & \hfill c\hfill \\ \hfill -c\hfill & \hfill d\hfill & \hfill a\hfill & \hfill -b\hfill \\ \hfill -d\hfill & \hfill -c\hfill & \hfill b\hfill & \hfill a\hfill \end{array}\right)$

as $ℝ$-linear maps via left action. Lemma 7(b) implies that for all elements $x,y\in {\mathrm{U}}_{1}\left(ℍ\right)$ one has ${L}_{x}{R}_{y}={R}_{y}{L}_{x}$ and that up to scalar multiplication with -1 the matrices ${L}_{x}$ and ${R}_{y}$ are uniquely determined by their product.

• (c)

The action from Lemma 5 translates into

$\omega :\mathrm{SO}\left(4\right)×\mathcal{ℋ}\left(ℝ\right)\to \mathcal{ℋ}\left(ℝ\right),\left({L}_{a}{R}_{b},\left(x,y\right)\right)↦\left({L}_{a}{R}_{a}\cdot x,{L}_{a}{R}_{b}\cdot y\right).$

• (d)

For $\alpha \in ℝ$, one has ${\eta }_{p}\left(D\left(\alpha \right)\right)={L}_{x}\cdot {R}_{x}$ with $x=\mathrm{cos}\left(\frac{\alpha }{2}\right)+\mathrm{sin}\left(\frac{\alpha }{2}\right)i$, i.e.,

$\left(\begin{array}{cc}\hfill {I}_{2}\hfill & \hfill \hfill \\ \hfill \hfill & \hfill D\left(\alpha \right)\hfill \end{array}\right)=\left(\begin{array}{cc}\hfill D\left(\frac{\alpha }{2}\right)\hfill & \hfill \hfill \\ \hfill \hfill & \hfill D\left(\frac{\alpha }{2}\right)\hfill \end{array}\right)\left(\begin{array}{cc}\hfill D\left(-\frac{\alpha }{2}\right)\hfill & \hfill \hfill \\ \hfill \hfill & \hfill D\left(\frac{\alpha }{2}\right)\hfill \end{array}\right).$

• (e)

For $\alpha \in ℝ$, we have ${\eta }_{l}\left(D\left(\alpha \right)\right)={L}_{x}\cdot {R}_{y}$ with $x=\mathrm{cos}\left(\frac{\alpha }{2}\right)+\mathrm{sin}\left(\frac{\alpha }{2}\right)k$ and $y=\mathrm{cos}\left(\frac{3\alpha }{2}\right)-\mathrm{sin}\left(\frac{3\alpha }{2}\right)k$, i.e.,

${\eta }_{l}\left(D\left(\alpha \right)\right)=\left(\begin{array}{ccc}\hfill \mathrm{cos}\left(\frac{\alpha }{2}\right)\hfill & \hfill \hfill & \hfill -\mathrm{sin}\left(\frac{\alpha }{2}\right)\hfill \\ \hfill \hfill & \hfill D\left(\frac{\alpha }{2}\right)\hfill & \hfill \hfill \\ \hfill \mathrm{sin}\left(\frac{\alpha }{2}\right)\hfill & \hfill \hfill & \hfill \mathrm{cos}\left(\frac{\alpha }{2}\right)\hfill \end{array}\right)\left(\begin{array}{ccc}\hfill \mathrm{cos}\left(\frac{3\alpha }{2}\right)\hfill & \hfill \hfill & \hfill -\mathrm{sin}\left(\frac{3\alpha }{2}\right)\hfill \\ \hfill \hfill & \hfill D\left(-\frac{3\alpha }{2}\right)\hfill & \hfill \hfill \\ \hfill \mathrm{sin}\left(\frac{3\alpha }{2}\right)\hfill & \hfill \hfill & \hfill \mathrm{cos}\left(\frac{3\alpha }{2}\right)\hfill \end{array}\right).$

The subgroup of right translations ${R}_{b}$ is normal in $\mathrm{SO}\left(4\right)$, the resulting quotient is isomorphic to $\mathrm{SO}\left(3\right)$. This canonical projection induces a surjection from the split Cayley hexagon onto the real projective plane, given by the projection ${〈\left({x}_{1},{x}_{2}\right)〉}_{ℝ}↦{〈{x}_{1}〉}_{ℝ}$. (Cf. [16, Section 5]. An alternative description of this surjection can be found in [18].)

The following lemma describes how the corresponding embedded circle groups behave under this surjection.

The following statements hold:

• (a)

There is an epimorphism ${\eta }_{1}:\mathrm{SO}\left(4\right)\to \mathrm{SO}\left(3\right)$ such that

${\eta }_{1}\circ {\eta }_{p}={\epsilon }_{23},{\eta }_{1}\circ {\eta }_{l}={\epsilon }_{12}.$

• (b)

There is an epimorphism ${\eta }_{2}:\mathrm{SO}\left(4\right)\to \mathrm{SO}\left(3\right)$ such that

${\eta }_{2}\circ {\eta }_{p}={\epsilon }_{12},{\eta }_{2}\circ {\eta }_{l}={\epsilon }_{23}.$

Proof.

By Remark 3, the map

$\psi :\mathrm{SO}\left(4\right)\to {\epsilon }_{\left\{2,3,4\right\}}\left(\mathrm{SO}\left(3\right)\right),{L}_{a}{R}_{b}↦{L}_{a}{R}_{a}$

$\psi \circ {\eta }_{p}={\eta }_{p}={\epsilon }_{34},$

and, by Remark (e),

$\left(\psi \circ {\eta }_{l}\right)\left(D\left(\alpha \right)\right)=\left(\begin{array}{ccc}\hfill \mathrm{cos}\left(\frac{\alpha }{2}\right)\hfill & \hfill \hfill & \hfill -\mathrm{sin}\left(\frac{\alpha }{2}\right)\hfill \\ \hfill \hfill & \hfill D\left(\frac{\alpha }{2}\right)\hfill & \hfill \hfill \\ \hfill \mathrm{sin}\left(\frac{\alpha }{2}\right)\hfill & \hfill \hfill & \hfill \mathrm{cos}\left(\frac{\alpha }{2}\right)\hfill \end{array}\right)\left(\begin{array}{ccc}\hfill \mathrm{cos}\left(\frac{\alpha }{2}\right)\hfill & \hfill \hfill & \hfill \mathrm{sin}\left(\frac{\alpha }{2}\right)\hfill \\ \hfill \hfill & \hfill D\left(\frac{\alpha }{2}\right)\hfill & \hfill \hfill \\ \hfill -\mathrm{sin}\left(\frac{\alpha }{2}\right)\hfill & \hfill \hfill & \hfill \mathrm{cos}\left(\frac{\alpha }{2}\right)\hfill \end{array}\right)$$=\left(\begin{array}{ccc}\hfill 1\hfill & \hfill \hfill & \hfill \hfill \\ \hfill \hfill & \hfill D\left(\alpha \right)\hfill & \hfill \hfill \\ \hfill \hfill & \hfill \hfill & \hfill 1\hfill \end{array}\right)={\epsilon }_{23}\left(D\left(\alpha \right)\right).$

Therefore, the map ${\eta }_{1}:={\epsilon }_{\left\{2,3,4\right\}}^{-1}\circ \psi$ has the desired properties. The existence of the map ${\eta }_{2}$ now follows from Lemma 4. ∎

The final result of this section allows us to carry out Strategy 1 for edges of type ${\mathrm{G}}_{2}$ in Theorem 11 below.

The following statements hold:

• (a)

There is an epimorphism ${\stackrel{~}{\eta }}_{1}:\mathrm{Spin}\left(4\right)\to \mathrm{Spin}\left(3\right)$ such that

${\stackrel{~}{\eta }}_{1}\circ {\stackrel{~}{\eta }}_{p}={\stackrel{~}{\epsilon }}_{23},{\stackrel{~}{\eta }}_{1}\circ {\stackrel{~}{\eta }}_{l}={\stackrel{~}{\epsilon }}_{12}.$

• (b)

There is an epimorphism ${\stackrel{~}{\eta }}_{2}:\mathrm{Spin}\left(4\right)\to \mathrm{Spin}\left(3\right)$ such that

${\stackrel{~}{\eta }}_{2}\circ {\stackrel{~}{\eta }}_{p}={\stackrel{~}{\epsilon }}_{12},{\stackrel{~}{\eta }}_{2}\circ {\stackrel{~}{\eta }}_{l}={\stackrel{~}{\epsilon }}_{23}.$

Proof.

The map ${\stackrel{~}{\eta }}_{1}:\mathrm{Spin}\left(4\right)\to \mathrm{Spin}\left(3\right)$, $u+𝕀v↦u+v$ makes the inner/right-hand quadrangle of the following diagram commute (see Remarks 2 and 4):

The lower triangle commutes by Lemma 14. The left-hand quadrangle commutes by Lemma 11. Hence

${\epsilon }_{23}\circ {\rho }_{2}=\eta \circ {\eta }_{p}\circ {\rho }_{2}=\eta \circ {\rho }_{4}\circ {\stackrel{~}{\eta }}_{p}={\rho }_{3}\circ {\stackrel{~}{\eta }}_{1}\circ {\stackrel{~}{\eta }}_{p}$

and

$\mathrm{ker}\left({\rho }_{3}\circ {\stackrel{~}{\eta }}_{1}\circ {\stackrel{~}{\eta }}_{p}\right)=\mathrm{ker}\left({\epsilon }_{23}\circ {\rho }_{2}\right)=\mathrm{ker}{\rho }_{2}=\left\{±{1}_{\mathrm{Spin}\left(2\right)}\right\}.$

Therefore ${\stackrel{~}{\epsilon }}_{23}={\stackrel{~}{\eta }}_{1}\circ {\stackrel{~}{\eta }}_{p}$ by Proposition 5. In particular, also the upper triangle of the diagram commutes. The second claim concerning ${\stackrel{~}{\eta }}_{1}$ follows by an analogous argument The claims concerning ${\stackrel{~}{\eta }}_{2}$ are now immediate by Lemma 4 and Proposition 4

14 Diagrams of type ${\mathrm{C}}_{2}$

In this section we prepare Strategy 1 for diagrams of type ${\mathrm{C}}_{2}$.

Let ${\mathrm{Sp}}_{4}\left(ℝ\right)$ be the matrix group with respect to the $ℝ$-basis ${e}_{1}$, $i{e}_{1}$, ${e}_{2}$, $i{e}_{2}$ of ${ℂ}^{2}$ leaving the real alternating form

$\left(\left({x}_{1},{x}_{2},{x}_{3},{x}_{4}\right),\left({y}_{1},{y}_{2},{y}_{3},{y}_{4}\right)\right)↦{x}_{1}{y}_{2}-{x}_{2}{y}_{1}+{x}_{3}{y}_{4}-{x}_{4}{y}_{3}$

invariant.

The maximal compact subgroup of ${\mathrm{Sp}}_{4}\left(ℝ\right)$ is the group ${\mathrm{U}}_{2}\left(ℂ\right)$, embedded as follows. Let ${e}_{1}$, ${e}_{2}$ be the standard basis of ${ℂ}^{2}$ and consider ${\mathrm{U}}_{2}\left(ℂ\right)$ as the isometry group of the scalar product $\left(\left({v}_{1},{v}_{2}\right),\left({w}_{1},{w}_{2}\right)\right)↦\overline{{v}_{1}}{w}_{1}+\overline{{v}_{2}}{w}_{2}$. Defining

${x}_{1}:=Re\left({v}_{1}\right),$${x}_{2}:=Im\left({v}_{1}\right),$${x}_{3}\mathit{ }:=Re\left({v}_{2}\right),$${x}_{4}:=Im\left({v}_{2}\right),$${y}_{1}:=Re\left({w}_{1}\right),$${y}_{2}:=Im\left({w}_{1}\right),$${y}_{3}\mathit{ }:=Re\left({w}_{2}\right),$${y}_{4}:=Im\left({w}_{2}\right),$

we compute

$\overline{{v}_{1}}{w}_{1}+\overline{{v}_{2}}{w}_{2}={x}_{1}{y}_{1}+{x}_{2}{y}_{2}+{x}_{3}{y}_{3}+{x}_{4}{y}_{4}+i\left({x}_{1}{y}_{2}-{x}_{2}{y}_{1}+{x}_{3}{y}_{4}-{x}_{4}{y}_{3}\right).$

Since two complex numbers are equal if and only if real part and imaginary part coincide, the group ${\mathrm{U}}_{2}\left(ℂ\right)$ preserves the form ${x}_{1}{y}_{2}-{x}_{2}{y}_{1}+{x}_{3}{y}_{4}-{x}_{4}{y}_{3}$ and we have found an embedding in ${\mathrm{Sp}}_{4}\left(ℝ\right)$, acting naturally on the $ℝ$-vector space ${ℂ}^{2}$ with $ℝ$-basis ${e}_{1}$, $i{e}_{1}$, ${e}_{2}$, $i{e}_{2}$.

As ${\mathrm{U}}_{2}\left(ℂ\right)$ also preserves the form ${x}_{1}{y}_{1}+{x}_{2}{y}_{2}+{x}_{3}{y}_{3}+{x}_{4}{y}_{4}$ with respect to the $ℝ$-basis ${e}_{1}$, $i{e}_{1}$, ${e}_{2}$, $i{e}_{2}$ of ${ℂ}^{2}$ it is at the same time also a subgroup of $\mathrm{O}\left(4\right)$, in fact of $\mathrm{SO}\left(4\right)$, since ${\mathrm{U}}_{2}\left(ℂ\right)$ is connected with respect to its Lie group topology. We will give concrete coordinates for this embedding in Remark 7 below.

The real symplectic quadrangle can be modelled as the point-line geometry consisting of the one-dimensional and two-dimensional subspaces of ${ℝ}^{4}$ which are totally isotropic with respect to a nondegenerate alternating bilinear form, cf. [44, Section 2.3.17], also [1, Section 10.1].

Using the $ℝ$-basis ${e}_{1}$, $i{e}_{1}$, ${e}_{2}$, $i{e}_{2}$ and the $ℝ$-alternating form on ${ℂ}^{2}$ given in Remark 2, the flag ${〈{e}_{1}〉}_{ℝ}\subset {〈{e}_{1},{e}_{2}〉}_{ℝ}$ is an incident point-line pair of the resulting symplectic quadrangle of ${\mathrm{Sp}}_{4}\left(ℝ\right)$. The stabiliser in ${\mathrm{U}}_{2}\left(ℂ\right)$ of the point ${〈{e}_{1}〉}_{ℝ}$ is isomorphic to ${\mathrm{O}}_{1}\left(ℝ\right)×{\mathrm{U}}_{1}\left(ℂ\right)$ where the first factor acts diagonally on ${〈{e}_{1}〉}_{ℝ}\oplus {〈i{e}_{1}〉}_{ℝ}$ and the second factor acts naturally on ${〈{e}_{2}〉}_{ℂ}$. The stabiliser of the line ${〈{e}_{1},{e}_{2}〉}_{ℝ}$ is isomorphic to ${\mathrm{O}}_{2}\left(ℝ\right)$ acting diagonally on ${〈{e}_{1},{e}_{2}〉}_{ℝ}\oplus {〈i{e}_{1},i{e}_{2}〉}_{ℝ}$.

Recall the definition of $D\left(\alpha \right)$ from Notation 1. Let

${\zeta }_{p}:\mathrm{SO}\left(2\right)\to {\mathrm{U}}_{2}\left(ℂ\right)\subset {\mathrm{Sp}}_{4}\left(ℝ\right)\cap \mathrm{SO}\left(4\right),$$D\left(\alpha \right)↦\left(\begin{array}{c}\hfill 1\hfill \\ \hfill \hfill & \hfill 1\hfill \\ \hfill \hfill & \hfill \hfill & \hfill \mathrm{cos}\left(\alpha \right)\hfill & \hfill \mathrm{sin}\left(\alpha \right)\hfill \\ \hfill \hfill & \hfill \hfill & \hfill -\mathrm{sin}\left(\alpha \right)\hfill & \hfill \mathrm{cos}\left(\alpha \right)\hfill \end{array}\right)$

and let

${\zeta }_{l}:\mathrm{SO}\left(2\right)\to {\mathrm{U}}_{2}\left(ℂ\right)\subset {\mathrm{Sp}}_{4}\left(ℝ\right)\cap \mathrm{SO}\left(4\right),$$D\left(\alpha \right)↦\stackrel{~}{D}\left(-\alpha \right):=\left(\begin{array}{ccc}\hfill \mathrm{cos}\left(\alpha \right)\hfill & \hfill \hfill & \hfill \mathrm{sin}\left(\alpha \right)\hfill \\ \hfill \hfill & \hfill \mathrm{cos}\left(\alpha \right)\hfill & \hfill \hfill & \hfill \mathrm{sin}\left(\alpha \right)\hfill \\ \hfill -\mathrm{sin}\left(\alpha \right)\hfill & \hfill \hfill & \hfill \mathrm{cos}\left(\alpha \right)\hfill & \hfill \hfill \\ \hfill \hfill & \hfill -\mathrm{sin}\left(\alpha \right)\hfill & \hfill \hfill & \hfill \mathrm{cos}\left(\alpha \right)\hfill \end{array}\right)$

be the embeddings of the circle group arising as point-stabilizing resp. line-stabilizing rank one groups as above with respect to the $ℝ$-bases ${e}_{1}$, ${e}_{2}$ of ${ℝ}^{2}$ and ${e}_{1}$, $i{e}_{1}$, ${e}_{2}$, $i{e}_{2}$ of ${ℂ}^{2}$.

In the following, we shall identify $ℂ=\left\{x+iy\mid x,y\in ℝ\right\}$ with $\left\{\left(\begin{array}{cc}\hfill x\hfill & \hfill -y\hfill \\ \hfill y\hfill & \hfill x\hfill \end{array}\right)\mid x,y\in ℝ\right\}$. This identification, in particular, embeds $\mathrm{SO}\left(2\right)$ into $ℂ$ as the unit circle group. This induces an embedding ${\mathrm{GL}}_{2}\left(ℂ\right)\to {\mathrm{GL}}_{4}\left(ℝ\right)$, $g↦\stackrel{~}{g}$. Note that for $\alpha \in ℝ$, this embedding yields $\stackrel{~}{D\left(\alpha \right)}=\stackrel{~}{D}\left(\alpha \right)\in {\mathrm{U}}_{2}\left(ℂ\right)$.

Let $B\mathrm{:=}\mathrm{diag}\mathit{}\mathrm{\left(}\mathrm{-}\mathrm{1}\mathrm{,}\mathrm{1}\mathrm{,}\mathrm{-}\mathrm{1}\mathrm{,}\mathrm{1}\mathrm{\right)}$, $C\mathrm{:=}\mathrm{diag}\mathit{}\mathrm{\left(}\mathrm{-}\mathrm{1}\mathrm{,}\mathrm{-}\mathrm{1}\mathrm{,}\mathrm{1}\mathrm{,}\mathrm{1}\mathrm{\right)}\mathrm{\in }{\mathrm{U}}_{\mathrm{2}}\mathit{}\mathrm{\left(}\mathrm{C}\mathrm{\right)}$. Then the following hold:

• (a)

The map ${\gamma }_{B}:{\mathrm{U}}_{2}\left(ℂ\right)\to {\mathrm{U}}_{2}\left(ℂ\right):A↦B\cdot A\cdot {B}^{-1}=B\cdot A\cdot B$ is an automorphism of ${\mathrm{U}}_{2}\left(ℂ\right)$ such that

${\gamma }_{B}\circ {\zeta }_{p}={\zeta }_{p}\circ \mathrm{inv}\mathit{ }\text{𝑎𝑛𝑑}\mathit{ }{\gamma }_{B}\circ {\zeta }_{l}={\zeta }_{l}.$

• (b)

The map ${\gamma }_{C}:{\mathrm{U}}_{2}\left(ℂ\right)\to {\mathrm{U}}_{2}\left(ℂ\right):A↦C\cdot A\cdot {C}^{-1}=C\cdot A\cdot C$ is an automorphism of ${\mathrm{U}}_{2}\left(ℂ\right)$ such that

${\gamma }_{C}\circ {\zeta }_{p}={\zeta }_{p}\mathit{ }\text{𝑎𝑛𝑑}\mathit{ }{\gamma }_{C}\circ {\zeta }_{l}={\zeta }_{l}\circ \mathrm{inv}.$

Proof.

Straightforward. ∎

Let $\alpha \mathrm{\in }\mathrm{R}$. Then $\stackrel{\mathrm{~}}{D}\mathit{}\mathrm{\left(}\alpha \mathrm{\right)}\mathrm{\in }{\mathrm{SU}}_{\mathrm{2}}\mathit{}\mathrm{\left(}\mathrm{C}\mathrm{\right)}$ and $\mathrm{\left(}\begin{array}{cc}\hfill D\mathit{}\mathrm{\left(}\mathrm{-}\alpha \mathrm{\right)}\hfill & \hfill \hfill \\ \hfill \hfill & \hfill D\mathit{}\mathrm{\left(}\alpha \mathrm{\right)}\hfill \end{array}\mathrm{\right)}\mathrm{\in }{\mathrm{SU}}_{\mathrm{2}}\mathit{}\mathrm{\left(}\mathrm{C}\mathrm{\right)}$.

Proof.

The first claim follows from the discussion in Notation 4. For the second claim, we compute

$det\left(\begin{array}{cc}\hfill D\left(-\alpha \right)\hfill & \hfill \hfill \\ \hfill \hfill & \hfill D\left(\alpha \right)\hfill \end{array}\right)=D\left(-\alpha \right)\cdot D\left(\alpha \right)={I}_{2}={1}_{ℂ}.\mathit{∎}$

Returning to the embedding ${\mathrm{U}}_{2}\left(ℂ\right)\to \mathrm{SO}\left(4\right)$ mentioned in Remark 2, the group

$\begin{array}{cc}\hfill {\mathrm{SU}}_{2}\left(ℂ\right)& =\left\{\left(\begin{array}{cc}\hfill {x}_{1}+i{y}_{1}\hfill & \hfill {x}_{2}+i{y}_{2}\hfill \\ \hfill -{x}_{2}+i{y}_{2}\hfill & \hfill {x}_{1}-i{y}_{2}\hfill \end{array}\right)|{x}_{1},{x}_{2},{y}_{1},{y}_{2}\in ℝ,{x}_{1}^{2}+{x}_{2}^{2}+{y}_{1}^{2}+{y}_{2}^{2}=1\right\}\hfill \\ & \le {\mathrm{U}}_{2}\left(ℂ\right)\hfill \end{array}$

acts $ℝ$-linearly on ${ℂ}^{2}$ with transformation matrices

$\left(\begin{array}{cccc}\hfill {x}_{1}\hfill & \hfill -{y}_{1}\hfill & \hfill {x}_{2}\hfill & \hfill -{y}_{2}\hfill \\ \hfill {y}_{1}\hfill & \hfill {x}_{1}\hfill & \hfill {y}_{2}\hfill & \hfill {x}_{2}\hfill \\ \hfill -{x}_{2}\hfill & \hfill -{y}_{2}\hfill & \hfill {x}_{1}\hfill & \hfill {y}_{1}\hfill \\ \hfill {y}_{2}\hfill & \hfill -{x}_{2}\hfill & \hfill -{y}_{1}\hfill & \hfill {x}_{1}\hfill \end{array}\right)$

with respect to the basis ${e}_{1}$, $i{e}_{1}$, ${e}_{2}$, $i{e}_{2}$. Remark 13 implies that the map

${\mathrm{SU}}_{2}\left(ℂ\right)\to \mathrm{SO}\left(4\right)$

given by

$\left(\begin{array}{cc}\hfill {x}_{1}+i{y}_{1}\hfill & \hfill {x}_{2}+i{y}_{2}\hfill \\ \hfill -{x}_{2}+i{y}_{2}\hfill & \hfill {x}_{1}-i{y}_{2}\hfill \end{array}\right)↦{R}_{{x}_{1}-{y}_{1}i+{x}_{2}j-{y}_{2}k}=\left(\begin{array}{cccc}\hfill {x}_{1}\hfill & \hfill -{y}_{1}\hfill & \hfill {x}_{2}\hfill & \hfill -{y}_{2}\hfill \\ \hfill {y}_{1}\hfill & \hfill {x}_{1}\hfill & \hfill {y}_{2}\hfill & \hfill {x}_{2}\hfill \\ \hfill -{x}_{2}\hfill & \hfill -{y}_{2}\hfill & \hfill {x}_{1}\hfill & \hfill {y}_{1}\hfill \\ \hfill {y}_{2}\hfill & \hfill -{x}_{2}\hfill & \hfill -{y}_{1}\hfill & \hfill {x}_{1}\hfill \end{array}\right)$

injects ${\mathrm{SU}}_{2}\left(ℂ\right)$ into $\mathrm{SO}\left(4\right)$. The restriction of this map to $\mathrm{SO}\left(2\right)\subset {\mathrm{SU}}_{2}\left(ℂ\right)$ by setting ${y}_{1}=0={y}_{2}$ provides the transformations $\stackrel{~}{D}$ from Definition 3. The group

$\left\{\left(\begin{array}{cc}\hfill \mathrm{cos}\left(\alpha \right)+i\mathrm{sin}\left(\alpha \right)\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill \mathrm{cos}\left(\alpha \right)+i\mathrm{sin}\left(\alpha \right)\hfill \end{array}\right)\right\}\cong {\mathrm{U}}_{1}\left(ℂ\right)\cong \mathrm{SO}\left(2\right)$

acts with transformation matrices

$\left(\begin{array}{cc}\hfill D\left(\alpha \right)\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill D\left(\alpha \right)\hfill \end{array}\right)=\left(\begin{array}{cccc}\hfill \mathrm{cos}\left(\alpha \right)\hfill & \hfill -\mathrm{sin}\left(\alpha \right)\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill \mathrm{sin}\left(\alpha \right)\hfill & \hfill \mathrm{cos}\left(\alpha \right)\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill \mathrm{cos}\left(\alpha \right)\hfill & \hfill -\mathrm{sin}\left(\alpha \right)\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill \mathrm{sin}\left(\alpha \right)\hfill & \hfill \mathrm{cos}\left(\alpha \right)\hfill \end{array}\right),$

i.e., Remark 13 implies that the map

${\mathrm{U}}_{1}\left(ℂ\right)\to \mathrm{SO}\left(4\right)$

given by

$\begin{array}{cc}& \left(\begin{array}{cc}\hfill \mathrm{cos}\left(\alpha \right)+i\mathrm{sin}\left(\alpha \right)\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill \mathrm{cos}\left(\alpha \right)+i\mathrm{sin}\left(\alpha \right)\hfill \end{array}\right)\hfill \\ & ↦{L}_{\mathrm{cos}\left(\alpha \right)+i\mathrm{sin}\left(\alpha \right)}=\left(\begin{array}{cccc}\hfill \mathrm{cos}\left(\alpha \right)\hfill & \hfill -\mathrm{sin}\left(\alpha \right)\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill \mathrm{sin}\left(\alpha \right)\hfill & \hfill \mathrm{cos}\left(\alpha \right)\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill \mathrm{cos}\left(\alpha \right)\hfill & \hfill -\mathrm{sin}\left(\alpha \right)\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill \mathrm{sin}\left(\alpha \right)\hfill & \hfill \mathrm{cos}\left(\alpha \right)\hfill \end{array}\right)\hfill \end{array}$

injects ${\mathrm{U}}_{1}\left(ℂ\right)$ into $\mathrm{SO}\left(4\right)$. Altogether, using Remark 2, we obtain the following commutative diagram:

Our candidate for a spin cover of the group ${\mathrm{U}}_{2}\left(ℂ\right)$ therefore is its double cover

${\mathrm{U}}_{1}\left(ℂ\right)×{\mathrm{SU}}_{2}\left(ℂ\right)\to {\mathrm{U}}_{2}\left(ℂ\right),\left(z,A\right)↦zA.$

Note that the fundamental group of ${\mathrm{U}}_{2}\left(ℂ\right)$ equals $ℤ$, as the determinant map $det:{\mathrm{U}}_{2}\left(ℂ\right)\to {\mathrm{U}}_{1}\left(ℂ\right)\cong \mathrm{SO}\left(2\right)$ induces an isomorphism of fundamental groups; its simply connected universal cover is isomorphic to $ℝ×{\mathrm{SU}}_{2}\left(ℂ\right)$. The above double cover is unique up to isomorphism, because $ℤ$ has a unique subgroup of index two (cf. [22, Theorem 1.38]).

In the following, let

${\stackrel{~}{\zeta }}_{p}:\mathrm{Spin}\left(2\right)\to \mathrm{SO}\left(2\right)×{\mathrm{SU}}_{2}\left(ℂ\right)\subset \mathrm{Spin}\left(4\right),$$S\left(\alpha \right)↦\left(D\left(-\alpha \right),\left(\begin{array}{cc}\hfill D\left(\alpha \right)\hfill & \hfill \hfill \\ \hfill \hfill & \hfill D\left(-\alpha \right)\hfill \end{array}\right)\right),$${\stackrel{~}{\zeta }}_{l}:\mathrm{Spin}\left(2\right)\to \mathrm{SO}\left(2\right)×{\mathrm{SU}}_{2}\left(ℂ\right)\subset \mathrm{Spin}\left(4\right),$$S\left(\alpha \right)↦\left(id,\stackrel{~}{D}\left(-\alpha \right)\right)$

and let

$\stackrel{^}{\rho }:\mathrm{SO}\left(2\right)×{\mathrm{SU}}_{2}\left(ℂ\right)\to {\mathrm{U}}_{2}\left(ℂ\right),\left(z,A\right)↦\left(\begin{array}{cc}\hfill z\hfill & \hfill \hfill \\ \hfill \hfill & \hfill z\hfill \end{array}\right)\cdot A.$

Recall the maps

${\rho }_{2}:\mathrm{Spin}\left(2\right)\to \mathrm{SO}\left(2\right),$$S\left(\alpha \right)↦D\left(2\alpha \right),$$\mathrm{sq}:G\to G,$$x↦{x}^{2},$$\mathrm{inv}:G\to G,$$x↦{x}^{-1}.$

One has

${\stackrel{~}{\zeta }}_{p}={\stackrel{~}{\epsilon }}_{34},{\stackrel{~}{\zeta }}_{l}\circ \mathrm{sq}={\stackrel{~}{\epsilon }}_{23}\cdot {\stackrel{~}{\epsilon }}_{14}$

Proof.

Using the identification $\mathrm{Spin}\left(4\right)\cong \mathrm{Spin}\left(3\right)×\mathrm{Spin}\left(3\right)$ and considering the left-hand factor as transformations by left multiplication and the right-hand factor as transformations by right multiplication of unit quaternions we compute

${\stackrel{~}{\epsilon }}_{34}\left(\mathrm{cos}\left(\alpha \right)+\mathrm{sin}\left(\alpha \right){e}_{1}{e}_{2}\right)$$=\mathrm{cos}\left(\alpha \right)+\mathrm{sin}\left(\alpha \right){e}_{3}{e}_{4}$$=\mathrm{cos}\left(\alpha \right)-𝕀i\mathrm{sin}\left(\alpha \right)$$\text{by Remark 7.2}$$=\left(\mathrm{cos}\left(\alpha \right)-i\mathrm{sin}\left(\alpha \right),\mathrm{cos}\left(\alpha \right)+i\mathrm{sin}\left(\alpha \right)\right)$$\text{by Remark 7.2}$$=\left({L}_{\mathrm{cos}\left(\alpha \right)-i\mathrm{sin}\left(\alpha \right)},{R}_{\mathrm{cos}\left(\alpha \right)-i\mathrm{sin}\left(\alpha \right)}\right)$$\text{by Lemma 13.12, Remark 13.13}$$=\left(D\left(-\alpha \right),\left(\begin{array}{cc}\hfill D\left(\alpha \right)\hfill & \hfill \hfill \\ \hfill \hfill & \hfill D\left(-\alpha \right)\hfill \end{array}\right)\right)$$\text{by Remark 14.7}$$={\stackrel{~}{\zeta }}_{p}\left(\mathrm{cos}\left(\alpha \right)+\mathrm{sin}\left(\alpha \right){e}_{1}{e}_{2}\right)$

and

${\stackrel{~}{\epsilon }}_{23}\left(\mathrm{cos}\left(\alpha \right)+\mathrm{sin}\left(\alpha \right){e}_{1}{e}_{2}\right){\stackrel{~}{\epsilon }}_{14}\left(\mathrm{cos}\left(\alpha \right)+\mathrm{sin}\left(\alpha \right){e}_{1}{e}_{2}\right)$$=\left(\mathrm{cos}\left(\alpha \right)+\mathrm{sin}\left(\alpha \right){e}_{2}{e}_{3}\right)\left(\mathrm{cos}\left(\alpha \right)+\mathrm{sin}\left(\alpha \right){e}_{1}{e}_{4}\right)$$={\left(\mathrm{cos}\left(\alpha \right)\right)}^{2}+j\mathrm{cos}\left(\alpha \right)\mathrm{sin}\left(\alpha \right)$$+𝕀\left({\left(\mathrm{sin}\left(\alpha \right)\right)}^{2}-j\mathrm{cos}\left(\alpha \right)\mathrm{sin}\left(\alpha \right)\right)$$\text{by Remark 7.2}$$=\left(1,{\left(\mathrm{cos}\left(\alpha \right)+j\mathrm{sin}\left(\alpha \right)\right)}^{2}\right)$$\text{by Remark 7.2}$$=\left(id,{R}_{{\left(\mathrm{cos}\left(\alpha \right)+j\mathrm{sin}\left(\alpha \right)\right)}^{2}}\right)$$\text{by Lemma 13.12, Remark 13.13}$$=\left(id,\stackrel{~}{D}\left(-2\alpha \right)\right)$$\text{by Remark 14.7}$$=\left({\stackrel{~}{\zeta }}_{l}\circ \mathrm{sq}\right)\left(\mathrm{cos}\left(\alpha \right)+\mathrm{sin}\left(\alpha \right){e}_{1}{e}_{2}\right),$

as desired. ∎

The following observation is the analog of Lemma 11.

One has

$\stackrel{^}{\rho }\circ {\stackrel{~}{\zeta }}_{p}={\zeta }_{p}\circ {\rho }_{2},\stackrel{^}{\rho }\circ {\stackrel{~}{\zeta }}_{l}\circ \mathrm{sq}=\stackrel{^}{\rho }\circ \mathrm{sq}\circ {\stackrel{~}{\zeta }}_{l}={\zeta }_{l}\circ {\rho }_{2}.$

Moreover,

${\left(\stackrel{^}{\rho }\right)}^{-1}\left({\zeta }_{p}\right)\left(\mathrm{SO}\left(2\right)\right)\cong \mathrm{Spin}\left(2\right),$${\left(\stackrel{^}{\rho }\right)}^{-1}\left({\zeta }_{l}\right)\left(\mathrm{SO}\left(2\right)\right)\cong \left\{1,-1\right\}×\mathrm{SO}\left(2\right).$

Proof.

For $\alpha \in ℝ$,

$\begin{array}{cc}\hfill \left(\stackrel{^}{\rho }\circ {\stackrel{~}{\zeta }}_{p}\right)\left(S\left(\alpha \right)\right)& =\stackrel{^}{\rho }\left(D\left(-\alpha \right),\left(\begin{array}{cc}\hfill D\left(\alpha \right)\hfill & \hfill \hfill \\ \hfill \hfill & \hfill D\left(-\alpha \right)\hfill \end{array}\right)\right)\hfill \\ & =\left(\begin{array}{cc}\hfill {1}_{\mathrm{SO}\left(2\right)}\hfill & \hfill \hfill \\ \hfill \hfill & \hfill D\left(-2\alpha \right)\hfill \end{array}\right)\hfill \\ & ={\zeta }_{p}\left(D\left(2\alpha \right)\right)\hfill \\ & =\left({\zeta }_{p}\circ {\rho }_{2}\right)\left(S\left(\alpha \right)\right)\hfill \end{array}$

and

$\begin{array}{cc}\hfill \left(\stackrel{^}{\rho }\circ \mathrm{sq}\circ {\stackrel{~}{\zeta }}_{l}\right)\left(S\left(\alpha \right)\right)& =\stackrel{^}{\rho }\left({1}_{\mathrm{SO}\left(2\right)},\stackrel{~}{D}\left(-2\alpha \right)\right)\hfill \\ & =\stackrel{~}{D}\left(-2\alpha \right)\hfill \\ & ={\zeta }_{l}\left(D\left(2\alpha \right)\right)\hfill \\ & =\left({\zeta }_{l}\circ {\rho }_{2}\right)\left(S\left(\alpha \right)\right).\hfill \end{array}$

For the second claim observe that

${\left(\stackrel{^}{\rho }\right)}^{-1}\left({\zeta }_{p}\right)\left(\mathrm{SO}\left(2\right)\right)=\left\{\left(D\left(-\alpha \right),\left(\begin{array}{cc}\hfill D\left(\alpha \right)\hfill & \hfill \hfill \\ \hfill \hfill & \hfill D\left(-\alpha \right)\hfill \end{array}\right)\right)|\alpha \in ℝ\right\}\cong \mathrm{Spin}\left(2\right),$${\left(\stackrel{^}{\rho }\right)}^{-1}\left({\zeta }_{l}\right)\left(\mathrm{SO}\left(2\right)\right)=\left\{\left(±id,\stackrel{~}{D}\left(-\alpha \right)\right)\mid \alpha \in ℝ\right\}\cong \left\{1,-1\right\}×\mathrm{SO}\left(2\right).\mathit{∎}$

One has

${\zeta }_{p}={\epsilon }_{34},{\zeta }_{l}={\epsilon }_{23}\cdot {\epsilon }_{14}$

Proof.

By Remark 13, Remark 7 and Lemma 10 this is immediate from Lemma 9. ∎

Given an automorphism $\tau \mathrm{\in }\mathrm{Aut}\mathit{}\mathrm{\left(}{\mathrm{U}}_{\mathrm{2}}\mathit{}\mathrm{\left(}\mathrm{C}\mathrm{\right)}\mathrm{\right)}$, there is a unique $\stackrel{\mathrm{~}}{\tau }\mathrm{\in }\mathrm{Aut}\mathit{}\mathrm{\left(}\mathrm{SO}\mathit{}\mathrm{\left(}\mathrm{2}\mathrm{\right)}\mathrm{×}{\mathrm{SU}}_{\mathrm{2}}\mathit{}\mathrm{\left(}\mathrm{C}\mathrm{\right)}\mathrm{\right)}$ such that

$\stackrel{^}{\rho }\circ \stackrel{~}{\tau }=\tau \circ \stackrel{^}{\rho }.$

Proof.

Let $\tau \in Aut\left({\mathrm{U}}_{2}\left(ℂ\right)\right)$. Consider the characteristic subgroups

${H}_{1}:=Z\left({\mathrm{U}}_{2}\left(ℂ\right)\right)=\left\{z\cdot {I}_{2}\mid z\in {\mathrm{U}}_{1}\left(ℂ\right)\cong \mathrm{SO}\left(2\right)\right\},$${H}_{2}:=\left[{\mathrm{U}}_{2}\left(ℂ\right),{\mathrm{U}}_{2}\left(ℂ\right)\right]={\mathrm{SU}}_{2}\left(ℂ\right)$

of ${\mathrm{U}}_{2}\left(ℂ\right)$. Note that ${H}_{1}$ and ${H}_{2}$ are also characteristic in $H:={H}_{1}×{H}_{2}$, as ${H}_{1}=Z\left(H\right)$ and ${H}_{2}=\left[H,H\right]$. Hence $Aut\left(H\right)=Aut\left({H}_{1}\right)×Aut\left({H}_{2}\right)$.

Let ${\tau }_{1}:={\tau }_{|{H}_{1}}\in Aut\left({H}_{1}\right)$ and ${\tau }_{2}:={\tau }_{|{H}_{2}}\in Aut\left({H}_{2}\right)$. Then the automorphism $\stackrel{~}{\tau }:=\left({\tau }_{1},{\tau }_{2}\right)\in Aut\left(H\right)$ satisfies

$\left(\stackrel{^}{\rho }\circ \stackrel{~}{\tau }\right)\left(z,A\right)=\stackrel{^}{\rho }\left({\tau }_{1}\left(z\right),{\tau }_{2}\left(A\right)\right)$$={\tau }_{1}\left(z\right){\tau }_{2}\left(A\right)=\tau \left(z\right)\tau \left(A\right)$$=\tau \left(zA\right)=\left(\tau \circ \stackrel{^}{\rho }\right)\left(z,A\right)$

for all $\left(z,A\right)\in \mathrm{SO}\left(2\right)×{\mathrm{SU}}_{2}\left(ℂ\right)$. Let $\psi =\left({\psi }_{1},{\psi }_{2}\right)\in Aut\left(\mathrm{SO}\left(2\right)×{\mathrm{SU}}_{2}\left(ℂ\right)\right)$ be such that $\stackrel{^}{\rho }\circ \psi =\tau \circ \stackrel{^}{\rho }$. Given $z\in \mathrm{SO}\left(2\right)$ and $A\in {\mathrm{SU}}_{2}\left(ℂ\right)$, one has

$\begin{array}{cc}\hfill {\psi }_{1}\left(z\right)& ={\psi }_{1}\left(z\right){\psi }_{2}\left({I}_{2}\right)=\left(\stackrel{^}{\rho }\circ \psi \right)\left(z,{I}_{2}\right)\hfill \\ & =\left(\tau \circ \stackrel{^}{\rho }\right)\left(z,{I}_{2}\right)=\left(\stackrel{^}{\rho }\circ \stackrel{~}{\tau }\right)\left(z,{I}_{2}\right)\hfill \\ & ={\tau }_{1}\left(z\right){\tau }_{2}\left({I}_{2}\right)={\tau }_{1}\left(z\right)\hfill \end{array}$

and

${\psi }_{2}\left(A\right)=\left(\stackrel{^}{\rho }\circ \psi \right)\left({1}_{\mathrm{SO}\left(2\right)},A\right)=\left(\stackrel{^}{\rho }\circ \stackrel{~}{\tau }\right)\left({1}_{\mathrm{SO}\left(2\right)},A\right)={\tau }_{2}\left(A\right).\mathit{ }\mathit{∎}$

The final result of this section allows us to carry out Strategy 1 for edges of type ${\mathrm{C}}_{2}$ in Theorem 11 below.

There is an epimorphism $\stackrel{\mathrm{~}}{\zeta }\mathrm{:}\mathrm{SO}\mathit{}\mathrm{\left(}\mathrm{2}\mathrm{\right)}\mathrm{×}{\mathrm{SU}}_{\mathrm{2}}\mathit{}\mathrm{\left(}\mathrm{C}\mathrm{\right)}\mathrm{\to }\mathrm{Spin}\mathit{}\mathrm{\left(}\mathrm{3}\mathrm{\right)}$ such that

$\stackrel{~}{\zeta }\circ {\stackrel{~}{\zeta }}_{p}={\stackrel{~}{\epsilon }}_{12}\circ \mathrm{inv},\stackrel{~}{\zeta }\circ {\stackrel{~}{\zeta }}_{l}={\stackrel{~}{\epsilon }}_{23}.$

Proof.

The map

$\psi :\mathrm{Spin}\left(3\right)\to {\mathrm{SU}}_{2}\left(ℂ\right),$$a+b{e}_{1}{e}_{2}+c{e}_{2}{e}_{3}+d{e}_{3}{e}_{1}↦{R}_{a+bi+cj+dk}=\left(\begin{array}{cccc}\hfill a\hfill & \hfill b\hfill & \hfill c\hfill & \hfill d\hfill \\ \hfill -b\hfill & \hfill a\hfill & \hfill -d\hfill & \hfill c\hfill \\ \hfill -c\hfill & \hfill d\hfill & \hfill a\hfill & \hfill -b\hfill \\ \hfill -d\hfill & \hfill -c\hfill & \hfill b\hfill & \hfill a\hfill \end{array}\right)$

is a group isomorphism by Remarks 5 (b) and 13 (b) (see also [37, 11.26]), with the convention that the matrix group ${\mathrm{SU}}_{2}\left(ℂ\right)$ acts $ℝ$-linearly on ${ℂ}^{2}\cong ℍ$ with respect to the $ℝ$-basis 1, $i=i1$, j, $k=ij$ as in Definition 3 and Notation 4. Let

$\stackrel{~}{\zeta }:\mathrm{SO}\left(2\right)×{\mathrm{SU}}_{2}\left(ℂ\right)\to \mathrm{Spin}\left(3\right),\left(z,A\right)↦{\psi }^{-1}\left(A\right).$

For $\alpha \in ℝ$, one has

$\left(\stackrel{~}{\zeta }\circ {\stackrel{~}{\zeta }}_{p}\right)\left(S\left(\alpha \right)\right)=\stackrel{~}{\zeta }\left(D\left(-\alpha \right),\left(\begin{array}{cc}\hfill D\left(\alpha \right)\hfill & \hfill \hfill \\ \hfill \hfill & \hfill D\left(-\alpha \right)\hfill \end{array}\right)\right)$$=\mathrm{cos}\left(\alpha \right)-\mathrm{sin}\left(\alpha \right){e}_{1}{e}_{2}=\left({\stackrel{~}{\epsilon }}_{12}\circ \mathrm{inv}\right)\left(S\left(\alpha \right)\right)$

and

$\left(\stackrel{~}{\zeta }\circ {\stackrel{~}{\zeta }}_{l}\right)\left(S\left(\alpha \right)\right)=\stackrel{~}{\zeta }\left({1}_{\mathrm{SO}\left(2\right)},\stackrel{~}{D}\left(-\alpha \right)\right)=\mathrm{cos}\left(\alpha \right)+\mathrm{sin}\left(\alpha \right){e}_{2}{e}_{3}={\stackrel{~}{\epsilon }}_{23}\left(S\left(\alpha \right)\right).\mathit{∎}$

The following statements hold:

• (a)

There exist epimorphisms ${\zeta }_{1},{\zeta }_{2},{\zeta }_{3}:{\mathrm{U}}_{2}\left(ℂ\right)\to \mathrm{SO}\left(3\right)$ such that

${\zeta }_{1}\circ {\zeta }_{p}={\epsilon }_{12}\circ \mathrm{inv},$${\zeta }_{1}\circ {\zeta }_{l}={\epsilon }_{23},$${\zeta }_{2}\circ {\zeta }_{p}={\epsilon }_{12},$${\zeta }_{2}\circ {\zeta }_{l}={\epsilon }_{23},$${\zeta }_{3}\circ {\zeta }_{p}={\epsilon }_{23},$${\zeta }_{3}\circ {\zeta }_{l}={\epsilon }_{12}.$

• (b)

There exist epimorphisms ${\stackrel{~}{\zeta }}_{2},{\stackrel{~}{\zeta }}_{3}:\mathrm{SO}\left(2\right)×{\mathrm{SU}}_{2}\left(ℂ\right)\to \mathrm{Spin}\left(3\right)$ such that

${\stackrel{~}{\zeta }}_{2}\circ {\stackrel{~}{\zeta }}_{p}={\stackrel{~}{\epsilon }}_{12},$${\stackrel{~}{\zeta }}_{2}\circ {\stackrel{~}{\zeta }}_{l}={\stackrel{~}{\epsilon }}_{23},$${\stackrel{~}{\zeta }}_{3}\circ {\stackrel{~}{\zeta }}_{p}={\stackrel{~}{\epsilon }}_{23},$${\stackrel{~}{\zeta }}_{3}\circ {\stackrel{~}{\zeta }}_{l}={\stackrel{~}{\epsilon }}_{12}.$

Proof.

The kernel of the epimorphism $\mathrm{SO}\left(2\right)×{\mathrm{SU}}_{2}\left(ℂ\right)\to {\mathrm{U}}_{2}\left(ℂ\right)$, $\left(z,A\right)↦zA$ is equal to $〈\left(D\left(\pi \right),\stackrel{~}{D}\left(\pi \right)\right)〉$. One has $\stackrel{~}{\zeta }\left(D\left(\pi \right),\stackrel{~}{D}\left(\pi \right)\right)={\psi }^{-1}\left(-{I}_{4}\right)=-{1}_{\mathrm{Spin}\left(3\right)}$, whence $〈\left(D\left(\pi \right),\stackrel{~}{D}\left(\pi \right)\right)〉\subseteq \mathrm{ker}\left({\rho }_{3}\circ \stackrel{~}{\zeta }\right)$. Therefore, the claim concerning ${\zeta }_{1}$ follows from Proposition 13 and the homomorphism theorem of groups.

The claims about ${\zeta }_{2}$, ${\zeta }_{3}$ then follow from Lemma 7, resp. Lemma 4. A subsequent application of Proposition 4 yields the claims about ${\stackrel{~}{\zeta }}_{2}$, ${\stackrel{~}{\zeta }}_{3}$. ∎

15 Non-spherical diagrams of rank two

In this section we prepare Strategy 1 for non-spherical diagrams of rank two. For an introduction to the concept of a Kac–Moody root datum we refer the reader to [43, Introduction], [34, 7.1.1, p. 172], [31, Definition 5.1]. For the definition of simple connectedness see [34, 7.1.2]. Note that for a given generalised Cartan matrix, up to isomorphism, there exists a uniquely determined simply connected Kac–Moody root datum.

Let $r,s\in ℕ$ be such that $rs\ge 4$ and consider the generalised Cartan matrix of rank two given by

$A:={\left(a\left(i,j\right)\right)}_{i\in \left\{1,2\right\}}=\left(\begin{array}{cc}\hfill 2\hfill & \hfill -r\hfill \\ \hfill -s\hfill & \hfill 2\hfill \end{array}\right)$

and a simply connected Kac–Moody root datum

$\mathcal{𝒟}=\left(\left\{1,2\right\},A,\mathrm{\Lambda },{\left({c}_{i}\right)}_{i\in \left\{1,2\right\}},{\left({h}_{i}\right)}_{i\in \left\{1,2\right\}}\right).$

Let $G:=G\left(A\right):=G\left(\mathcal{𝒟}\right)$ be the corresponding (simply connected) real Kac–Moody group of rank two, let ${T}_{0}$ be the fundamental torus of G with respect to the fundamental roots ${\alpha }_{1}$, ${\alpha }_{2}$, and let

$K:={K}^{r,s}:=K\left(A\right)$

be the subgroup consisting of the elements fixed by the Cartan–Chevalley involution with respect to ${T}_{0}$ of G, i.e., K is the maximal compact subgroup of G.

Let ${G}_{i}\cong {\mathrm{SL}}_{2}\left(ℝ\right)$ be the corresponding fundamental subgroups of rank one and define

${K}_{i}:={K}_{i}^{r,s}:={G}_{i}\cap {K}^{r,s}\cong \mathrm{SO}\left(2\right)\mathit{ }\text{and}\mathit{ }T:={T}_{0}\cap K.$

By (KMG3) (see [43, p. 545] or, e.g., [31, p. 84]), the torus ${T}_{0}$ is generated by ${\mu }^{{h}_{i}}$ for $i=1,2$ and $\mu \in ℝ\\left\{0\right\}$ arbitrary. The action of the Cartan–Chevalley involution on the torus is given by ${\mu }^{{h}_{i}}↦{\left({\mu }^{-1}\right)}^{{h}_{i}}={\mu }^{-{h}_{i}}$. Hence

$T={T}_{0}\cap K$$=\left\{id={1}^{{h}_{1}}={1}^{{h}_{2}},{t}_{1}:={\left(-1\right)}^{{h}_{1}},{t}_{2}:={\left(-1\right)}^{{h}_{2}},{t}_{1}{t}_{2}={\left(-1\right)}^{{h}_{1}+{h}_{2}}\right\}$$\cong ℤ/2ℤ×ℤ/2ℤ.$

The group K is isomorphic to a free amalgamated product

${K}_{1}T{*}_{T}{K}_{2}T.$

Proof.

The twin building of the Kac–Moody group G is a twin tree (cf. [35, 36]). The chambers are the edges; the panels are the sets of edges sharing one vertex and, hence, correspond to vertices. The group K acts edge-transitively and without inversions on each half of the twin tree of G by the Iwasawa decomposition (see, e.g., [12]). The Cartan–Chevalley involution ω interchanges the two halves of the twin tree, mapping edges to opposite edges. Hence the stabiliser in K of the fundamental edge ${c}^{+}$ also stabilises the opposite edge ${c}^{-}=\omega \left({c}^{+}\right)$ and, thus, the unique twin apartment spanned by them. It follows that the edge stabiliser is T. Since the panels correspond to the vertices of the tree, the stabilisers of the vertices of the fundamental edge ${c}^{+}$ are equal to ${K}_{1}T$ and ${K}_{2}T$. The claim follows from [38, Chapter I, Section 5]. ∎

Since ${K}_{i}⊴{K}_{i}T$, this free amalgamated product is fully determined by the intersections ${K}_{i}\cap T$ and the action of T on each ${K}_{i}$. Note that (KMG3) implies $T\cap {K}_{1}=\left\{1,{t}_{1}\right\}$ and $T\cap {K}_{2}=\left\{1,{t}_{2}\right\}$. The action of T on each ${K}_{i}$ can be extracted from the action of ${T}_{0}$ on each ${G}_{i}$, which according to [43, (4), p. 549] (or also [31, (5.1), p. 86]) is given by

$t{x}_{i}\left(\lambda \right){t}^{-1}={x}_{i}\left(t\left({c}_{i}\right)\lambda \right)$

for $t\in {T}_{0}$, $\lambda \in ℝ$ and root group functor ${x}_{i}$. According to [43, Section 2, p. 544] (or also [31, Definitions 5.1 and 5.5]) one computes for $i,j\in \left\{1,2\right\}$ that

${t}_{i}\left({c}_{j}\right)={\left(-1\right)}^{{h}_{i}}\left({c}_{j}\right)={\left(-1\right)}^{{h}_{i}\left({c}_{j}\right)}={\left(-1\right)}^{a\left(i,j\right)}.$

We conclude that ${t}_{i}$ acts trivially on ${K}_{j}$, if and only if the entry $a\left(i,j\right)$ of the generalised Cartan matrix is even; conversely it acts non-trivially (and hence by inversion) if and only if $a\left(i,j\right)$ is odd.

In symbols, for

$n\left(i,j\right):=\left\{\begin{array}{cc}0,\hfill & \text{if}a\left(i,j\right)\text{is even},\hfill \\ 1,\hfill & \text{if}a\left(i,j\right)\text{is odd}.\hfill \end{array}$

and ${k}_{j}\in {K}_{j}$ one has

${t}_{i}^{-1}{k}_{j}{t}_{i}={k}_{j}^{-2n\left(i,j\right)}{k}_{j}.$(15.1)

Let

${\theta }_{1}:={\theta }_{1}^{r,s}:\mathrm{SO}\left(2\right)\to {K}_{1}^{r,s},{\theta }_{2}:={\theta }_{2}^{r,s}:\mathrm{SO}\left(2\right)\to {K}_{2}^{r,s},$

be continuous isomorphisms.

Let $r,s\in ℕ$ such that $rs\ge 4$, let $A=\left(\begin{array}{cc}\hfill 2\hfill & \hfill -r\hfill \\ \hfill -s\hfill & \hfill 2\hfill \end{array}\right)$. Then set

${H}^{r,s}:=\left\{\begin{array}{cc}\mathrm{SO}\left(2\right)×\mathrm{SO}\left(2\right),\hfill & \text{if}r\equiv s\equiv 0\left(mod2\right),\hfill \\ \mathrm{SO}\left(3\right),\hfill & \text{if}r\equiv s\equiv 1\left(mod2\right),\hfill \\ {\mathrm{U}}_{2}\left(ℂ\right),\hfill & \text{otherwise}.\hfill \end{array}$

Also set

$\begin{array}{cc}\hfill {\delta }_{1}& :=\left\{\begin{array}{cccc}& {\iota }_{1},\hfill & & \hfill \text{if}r\equiv s\equiv 0\left(mod2\right),\\ & {\epsilon }_{12},\hfill & & \hfill \text{if}r\equiv s\equiv 1\left(mod2\right),\\ & {\zeta }_{l},\hfill & & \hfill \text{if}r\equiv 0,s\equiv 1\left(mod2\right),\\ & {\zeta }_{p},\hfill & & \hfill \text{if}r\equiv 1,s\equiv 0\left(mod2\right),\end{array}\hfill \\ \hfill {\delta }_{2}& :=\left\{\begin{array}{cccccc}& {\iota }_{2},\hfill & & \hfill \text{if}r\equiv s\equiv 0\left(mod2\right)& & \text{(see Notation 8.1)},\hfill \\ & {\epsilon }_{23},\hfill & & \hfill \text{if}r\equiv s\equiv 1\left(mod2\right)& & \text{(see Notation 5.2)},\hfill \\ & {\zeta }_{p},\hfill & & \hfill \text{if}r\equiv 0,s\equiv 1\left(mod2\right)& & \text{(see Definition 14.3)},\hfill \\ & {\zeta }_{l},\hfill & & \hfill \text{if}r\equiv 1,s\equiv 0\left(mod2\right).\end{array}\hfill \end{array}$

Let $r\mathrm{,}s\mathrm{\in }\mathrm{N}$ such that $r\mathit{}s\mathrm{\ge }\mathrm{4}$, let $A\mathrm{=}\mathrm{\left(}\begin{array}{cc}\hfill \mathrm{2}\hfill & \hfill \mathrm{-}r\hfill \\ \hfill \mathrm{-}s\hfill & \hfill \mathrm{2}\hfill \end{array}\mathrm{\right)}$, let $G\mathit{}\mathrm{\left(}A\mathrm{\right)}$ be the corresponding simply connected real Kac–Moody group, and let ${K}^{r\mathrm{,}s}$ be its maximal compact subgroup. Then there exists a group epimorphism $\theta \mathrm{:}{K}^{r\mathrm{,}s}\mathrm{\to }{H}^{r\mathrm{,}s}$ such that

$\theta \circ {\theta }_{1}={\delta }_{1}\mathit{ }\text{𝑎𝑛𝑑}\mathit{ }\theta \circ {\theta }_{2}={\delta }_{2}.$

Proof.

By Lemma 3 the group K is isomorphic to ${K}_{1}T{*}_{T}{K}_{2}T$ with $T=\left\{1,{t}_{1},{t}_{2},{t}_{1}{t}_{2}\right\}\cong ℤ/2ℤ×ℤ/2ℤ$ and ${T}_{0}\cap {K}_{1}=\left\{1,{t}_{1}\right\}$, ${T}_{0}\cap {K}_{2}=\left\{1,{t}_{2}\right\}$; in particular, K is generated by ${K}_{1}\cong \mathrm{SO}\left(2\right)$ and ${K}_{2}\cong \mathrm{SO}\left(2\right)$. As K is a free amalgamated product it therefore suffices to define θ on each of the ${K}_{i}$ and to verify that the actions of the ${t}_{i}$ on the ${K}_{j}$ are compatible with the actions of the images of the ${t}_{i}$ on the images of the ${K}_{j}$. Define θ via

${\theta }_{|{K}_{1}}:{K}_{1}\to {\delta }_{1}\left(\mathrm{SO}\left(2\right)\right),x↦\left({\delta }_{1}\circ \theta _{1}{}^{-1}\right)\left(x\right)$

and

${\theta }_{|{K}_{2}}:{K}_{2}\to {\delta }_{2}\left(\mathrm{SO}\left(2\right)\right),x↦\left({\delta }_{2}\circ \theta _{2}{}^{-1}\right)\left(x\right).$

Then this is compatible with the action of T. Indeed, using Remark 4, one observes:

• (a)

Since $r\equiv s\equiv 0\left(mod2\right)$, the elements ${t}_{i}$ centralise the groups ${K}_{j}$, which is compatible with the fact that $\mathrm{SO}\left(2\right)×\mathrm{SO}\left(2\right)$ is an abelian group.

• (b)

Since $r\equiv s\equiv 1\left(mod2\right)$, the element ${t}_{1}$ inverts the group ${K}_{2}$ and the element ${t}_{2}$ inverts the group ${K}_{1}$, which is compatible with the situation in $\mathrm{SO}\left(3\right)$ by Lemma 7.

• (c)

Since $r\equiv 0\left(mod2\right)$, $s\equiv 1\left(mod2\right)$, the element ${t}_{1}$ centralises ${K}_{2}$ and the element ${t}_{2}$ inverts the group ${K}_{1}$. This is compatible with the following computations (cf. Definition 3): for all $g\in {K}_{2}$,

$\begin{array}{cc}\hfill \theta \left({t}_{1}g{t}_{1}\right)& =\left(\begin{array}{ccc}\hfill -1\hfill & \hfill \hfill & \hfill 0\hfill \\ \hfill \hfill & \hfill -1\hfill & \hfill \hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill \hfill & \hfill -1\hfill & \hfill \hfill \\ \hfill \hfill & \hfill 0\hfill & \hfill \hfill & \hfill -1\hfill \end{array}\right)\left(\begin{array}{c}\hfill 1\hfill \\ \hfill \hfill & \hfill 1\hfill \\ \hfill \hfill & \hfill \hfill & \hfill x\hfill & \hfill y\hfill \\ \hfill \hfill & \hfill \hfill & \hfill -y\hfill & \hfill x\hfill \end{array}\right)\left(\begin{array}{ccc}\hfill -1\hfill & \hfill \hfill & \hfill 0\hfill \\ \hfill \hfill & \hfill -1\hfill & \hfill \hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill \hfill & \hfill -1\hfill & \hfill \hfill \\ \hfill \hfill & \hfill 0\hfill & \hfill \hfill & \hfill -1\hfill \end{array}\right)\hfill \\ & =\left(\begin{array}{c}\hfill 1\hfill \\ \hfill \hfill & \hfill 1\hfill \\ \hfill \hfill & \hfill \hfill & \hfill x\hfill & \hfill y\hfill \\ \hfill \hfill & \hfill \hfill & \hfill -y\hfill & \hfill x\hfill \end{array}\right)=\theta \left(g\right),\hfill \end{array}$

and for all $g\in {K}_{1}$,

$\begin{array}{cc}\hfill \theta \left({t}_{2}g{t}_{2}\right)& =\left(\begin{array}{c}\hfill 1\hfill \\ \hfill \hfill & \hfill 1\hfill \\ \hfill \hfill & \hfill \hfill & \hfill -1\hfill & \hfill 0\hfill \\ \hfill \hfill & \hfill \hfill & \hfill 0\hfill & \hfill -1\hfill \end{array}\right)\left(\begin{array}{ccc}\hfill x\hfill & \hfill \hfill & \hfill y\hfill \\ \hfill \hfill & \hfill x\hfill & \hfill \hfill & \hfill y\hfill \\ \hfill -y\hfill & \hfill \hfill & \hfill x\hfill & \hfill \hfill \\ \hfill \hfill & \hfill -y\hfill & \hfill \hfill & \hfill x\hfill \end{array}\right)\left(\begin{array}{c}\hfill 1\hfill \\ \hfill \hfill & \hfill 1\hfill \\ \hfill \hfill & \hfill \hfill & \hfill -1\hfill & \hfill 0\hfill \\ \hfill \hfill & \hfill \hfill & \hfill 0\hfill & \hfill -1\hfill \end{array}\right)\hfill \\ & =\left(\begin{array}{ccc}\hfill x\hfill & \hfill \hfill & \hfill -y\hfill \\ \hfill \hfill & \hfill x\hfill & \hfill \hfill & \hfill -y\hfill \\ \hfill y\hfill & \hfill \hfill & \hfill x\hfill & \hfill \hfill \\ \hfill \hfill & \hfill y\hfill & \hfill \hfill & \hfill x\hfill \end{array}\right)=\theta \left({g}^{-1}\right).\hfill \end{array}$

• (d)

This is dual to (c). ∎

Let $r,s\in ℕ$ such that $rs\ge 4$, let $A=\left(\begin{array}{cc}\hfill 2\hfill & \hfill -r\hfill \\ \hfill -s\hfill & \hfill 2\hfill \end{array}\right)$. Then set

${\stackrel{~}{H}}^{r,s}:=\left\{\begin{array}{cc}\mathrm{Spin}\left(2\right)×\mathrm{Spin}\left(2\right),\hfill & \text{if}r\equiv s\equiv 0\left(mod2\right),\hfill \\ \mathrm{Spin}\left(3\right),\hfill & \text{if}r\equiv s\equiv 1\left(mod2\right),\hfill \\ \mathrm{SO}\left(2\right)×{\mathrm{SU}}_{2}\left(ℂ\right),\hfill & \text{otherwise}.\hfill \end{array}$

Furthermore, set

$\begin{array}{cc}\hfill {\stackrel{~}{\delta }}_{1}& :=\left\{\begin{array}{cccc}& {\stackrel{~}{\iota }}_{1},\hfill & & \hfill \text{if}r\equiv s\equiv 0\left(mod2\right),\\ & {\stackrel{~}{\epsilon }}_{12},\hfill & & \hfill \text{if}r\equiv s\equiv 1\left(mod2\right),\\ & {\stackrel{~}{\zeta }}_{l},\hfill & & \hfill \text{if}r\equiv 0,s\equiv 1\left(mod2\right),\\ & {\stackrel{~}{\zeta }}_{p},\hfill & & \hfill \text{if}r\equiv 1,s\equiv 0\left(mod2\right),\end{array}\hfill \\ \hfill {\stackrel{~}{\delta }}_{2}& :=\left\{\begin{array}{cccccc}& {\stackrel{~}{\iota }}_{2},\hfill & & \hfill \text{if}r\equiv s\equiv 0\left(mod2\right)& & \text{(see Notation 8.1)},\hfill \\ & {\stackrel{~}{\epsilon }}_{23},\hfill & & \hfill \text{if}r\equiv s\equiv 1\left(mod2\right)& & \text{(see Lemma 6.10)},\hfill \\ & {\stackrel{~}{\zeta }}_{p},\hfill & & \hfill \text{if}r\equiv 0,s\equiv 1\left(mod2\right)& & \text{(see Notation 14.8)},\hfill \\ & {\stackrel{~}{\zeta }}_{l},\hfill & & \hfill \text{if}r\equiv 1,s\equiv 0\left(mod2\right).\end{array}\hfill \end{array}$

The central extension

$\overline{\rho }:{\stackrel{~}{H}}^{r,s}\to {H}^{r,s}$

satisfies

$\overline{\rho }=\left\{\begin{array}{cccccc}& {\rho }_{2}×{\rho }_{2},\hfill & & \hfill \text{if}r\equiv s\equiv 0\left(mod2\right)& & \text{(see Notation 8.1)},\hfill \\ & {\rho }_{3},\hfill & & \hfill \text{if}r\equiv s\equiv 1\left(mod2\right)& & \text{(see Theorem 6.8)},\hfill \\ & \stackrel{^}{\rho },\hfill & & \hfill \text{otherwise}& & \text{(see Notation 14.8)}.\hfill \end{array}$

Let ${K}^{r,s}=K\left(A\right)={K}_{1}T{*}_{T}{K}_{2}T$ be as in Lemma 3 and let ${t}_{1}\in {K}_{1}\cap T$, ${t}_{2}\in {K}_{2}\cap T$ as in Remark 4.

Define

${u}_{i}:=\theta \left({t}_{i}\right)\mathit{ }\text{(see Proposition 15.7)},U:=〈{u}_{1},{u}_{2}〉\cong ℤ/2ℤ×ℤ/2ℤ.$

Furthermore, define

$\stackrel{~}{U}:={\overline{\rho }}^{-1}\left(U\right),{\stackrel{~}{K}}_{i}:={\stackrel{~}{K}}_{i}^{r,s}:={\overline{\rho }}^{-1}\left(\theta \left({K}_{i}\right)\right),$

and the spin extension

$\stackrel{~}{K}:={\stackrel{~}{K}}^{r,s}:=\stackrel{~}{K}\left(A\right):={\stackrel{~}{K}}_{1}\stackrel{~}{U}{*}_{\stackrel{~}{U}}{\stackrel{~}{K}}_{2}\stackrel{~}{U}$

of

$K={K}^{r,s}=K\left(A\right),$

let

$\stackrel{^}{\stackrel{^}{\rho }}:{\stackrel{~}{K}}_{1}\stackrel{~}{U}{*}_{\stackrel{~}{U}}{\stackrel{~}{K}}_{2}\stackrel{~}{U}\to {K}_{1}T{*}_{T}{K}_{2}T$

be the epimorphism induced by ${\overline{\rho }}_{|{\stackrel{~}{K}}_{1}}$, ${\overline{\rho }}_{|{\stackrel{~}{K}}_{2}}$ and let

${\stackrel{~}{\theta }}_{1}:\mathrm{Spin}\left(2\right)\to {\stackrel{~}{K}}_{1}\mathit{ }\text{and}\mathit{ }{\stackrel{~}{\theta }}_{2}:\mathrm{Spin}\left(2\right)\to {\stackrel{~}{K}}_{2}$

be continuous monomorphisms such that the following diagrams commute for $i=1,2$:

One has ${\stackrel{~}{K}}_{1}\cong \mathrm{Spin}\left(2\right)$ unless $r\equiv 0,s\equiv 1\left(mod2\right)$ and ${\stackrel{~}{K}}_{2}\cong \mathrm{Spin}\left(2\right)$ unless $r\equiv 1,s\equiv 0\left(mod2\right)$, in which case the respective group is isomorphic to $\left\{1,-1\right\}×\mathrm{SO}\left(2\right)$ (cf. Lemma 10). Hence ${\stackrel{~}{\theta }}_{1}$ actually is a (continuous) isomorphism unless $r\equiv 0,s\equiv 1\left(mod2\right)$, in which case it is a (continuous) isomorphism onto the unique connected subgroup of index two of ${\stackrel{~}{K}}_{1}$.

For $i=1$ the map on the left-hand side of the above commutative diagram is

$\mathrm{Spin}\left(2\right)\to \mathrm{SO}\left(2\right),$$S\left(\alpha \right)↦D\left(\alpha \right)$$\text{if}r\equiv 0,s\equiv 1\left(mod2\right)$$\text{(see Notation 8.1)},$$\mathrm{Spin}\left(2\right)\to \mathrm{SO}\left(2\right),$$x↦{\rho }_{2}\left(x\right)$$\text{otherwise}.$

The dual statement holds for ${\stackrel{~}{\theta }}_{2}$.

In particular, for $i=2$ the map on the left-hand side of the above commutative diagram is

$\mathrm{Spin}\left(2\right)\to \mathrm{SO}\left(2\right),$$S\left(\alpha \right)↦D\left(\alpha \right)$$\text{if}r\equiv 1,s\equiv 0\left(mod2\right)$$\text{(see Notation 8.1)},$$\mathrm{Spin}\left(2\right)\to \mathrm{SO}\left(2\right),$$x↦{\rho }_{2}\left(x\right)$$\text{otherwise}.$

Define

${\stackrel{~}{t}}_{1}:=\left\{\begin{array}{cc}{\stackrel{~}{\theta }}_{1}\left(S\left(\pi \right)\right),\hfill & \text{if}r\equiv 0,s\equiv 1\left(mod2\right),\hfill \\ {\stackrel{~}{\theta }}_{1}\left(S\left(\frac{\pi }{2}\right)\right),\hfill & \text{otherwise},\hfill \end{array}$${\stackrel{~}{t}}_{2}:=\left\{\begin{array}{cc}{\stackrel{~}{\theta }}_{2}\left(S\left(\pi \right)\right),\hfill & \text{if}r\equiv 1,s\equiv 0\left(mod2\right),\hfill \\ {\stackrel{~}{\theta }}_{2}\left(S\left(\frac{\pi }{2}\right)\right),\hfill & \text{otherwise}.\hfill \end{array}$

The following is true by construction:

Let $r\mathrm{,}s\mathrm{\in }\mathrm{N}$ such that $r\mathit{}s\mathrm{\ge }\mathrm{4}$, let $A\mathrm{=}\mathrm{\left(}\begin{array}{cc}\hfill \mathrm{2}\hfill & \hfill \mathrm{-}r\hfill \\ \hfill \mathrm{-}s\hfill & \hfill \mathrm{2}\hfill \end{array}\mathrm{\right)}$, let $G\mathit{}\mathrm{\left(}A\mathrm{\right)}$ be the corresponding simply connected real Kac–Moody group, and let ${K}^{r\mathrm{,}s}$ be its maximal compact subgroup, and let ${\stackrel{\mathrm{~}}{K}}^{r\mathrm{,}s}$ be its spin extension. Then there exists a group epimorphism $\stackrel{\mathrm{~}}{\theta }\mathrm{:}{\stackrel{\mathrm{~}}{K}}^{r\mathrm{,}s}\mathrm{\to }{\stackrel{\mathrm{~}}{H}}^{r\mathrm{,}s}$ such that

$\stackrel{~}{\theta }\circ {\stackrel{~}{\theta }}_{1}={\stackrel{~}{\delta }}_{1}\mathit{ }\mathit{\text{and}}\mathit{ }\stackrel{~}{\theta }\circ {\stackrel{~}{\theta }}_{2}={\stackrel{~}{\delta }}_{2}.$

Moreover, the following diagram commutes:

where the epimorphism on the left-hand side is one of ${\rho }_{\mathrm{2}}$ or $S\mathit{}\mathrm{\left(}\alpha \mathrm{\right)}\mathrm{↦}D\mathit{}\mathrm{\left(}\alpha \mathrm{\right)}$ as described in Remark 9.

Furthermore, for $\mathrm{\left\{}i\mathrm{,}j\mathrm{\right\}}\mathrm{=}\mathrm{\left\{}\mathrm{1}\mathrm{,}\mathrm{2}\mathrm{\right\}}$ and ${\stackrel{\mathrm{~}}{k}}_{j}\mathrm{\in }{\stackrel{\mathrm{~}}{K}}_{j}$ one has

${\stackrel{~}{t}}_{i}^{-1}{\stackrel{~}{k}}_{j}{\stackrel{~}{t}}_{i}={\stackrel{~}{k}}_{j}^{-2n\left(i,j\right)}{\stackrel{~}{k}}_{j}.$(15.2)

Identity (15.2) follows from identity (15.1) in Remark 4

The Cartan matrix of type ${\mathrm{C}}_{2}$ over $\left\{1,2\right\}$ with short root ${\alpha }_{1}$ and long root ${\alpha }_{2}$ (i.e., $2\to 1$; see Remark 3) is

$\left(\begin{array}{cc}\hfill 2\hfill & \hfill -2\hfill \\ \hfill -1\hfill & \hfill 2\hfill \end{array}\right),$

cf. [6, p. 44]. The group ${H}^{r,s}$ from Proposition 10 is of type ${\mathrm{C}}_{2}$ with short root ${\alpha }_{1}$ and long root ${\alpha }_{2}$ if and only if r is even and s is odd. We conclude that for $i=1$ the map on the left-hand side of the commutative diagram is

$\mathrm{Spin}\left(2\right)\to \mathrm{SO}\left(2\right),S\left(\alpha \right)↦D\left(\alpha \right)$

which is in accordance with Notation 8 and Lemma 10. In other words, in this example the rank one group corresponding to the long root is doubly covered by its spin cover and the rank one group corresponding to the short root is singly covered by its spin cover.

The direction introduced for edges labelled $\mathrm{\infty }$ in Notation 4 was chosen to fit this observation: the arrow points away from the doubly covered vertex of the diagram towards the singly covered vertex of the diagram.

If both r, s are odd, then in $\stackrel{~}{K}={\stackrel{~}{K}}_{1}\stackrel{~}{U}{*}_{\stackrel{~}{U}}{\stackrel{~}{K}}_{2}\stackrel{~}{U}$ one has