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Journal of Group Theory

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Volume 20, Issue 3

Issues

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:
  • De Gruyter OnlineGoogle Scholar
/ 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 Wspin(E10) would be very useful for the study of fermionic billards. Lacking a concrete mathematical model of that group Wspin(E10), Damour and Hillmann in their article instead use images of Wspin(E10) 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 Wspin(E10) can be constructed as a discrete subgroup of a double spin cover Spin(E10) of the subgroup K(E10) of elements fixed by the Cartan–Chevalley involution of the split real Kac–Moody group of type E10. 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={1,,n} be a set of labels of the vertices of Π, and let K(Π) be the subgroup of elements fixed by the Cartan–Chevalley involution of the split real Kac–Moody group of type Π. For each iI let GiSpin(2) and for each ijI let

Gij{Spin(3),if i, j form an edge of Π,(Spin(2)×Spin(2))/(-1,-1),if i, j do not form an edge of Π.

Moreover, for i,jI with i<j, let ϕiji:GiGij be the standard embedding as “upper-left diagonal block” and let ϕijj:GjGij be the standard embedding as “lower-right diagonal block”.

Then up to isomorphism there exists a uniquely determined group, denoted Spin(Π), whose multiplication table extends the partial multiplication provided by (i<jIGij)/, where is the equivalence relation determined by

ϕiji(x)ϕiki(x)

for all ij,kI and xGi. Furthermore, there exists a canonical two-to-one central extension Spin(Π)K(Π).

The system {Gi,Gij,ϕiji} is called an amalgam of groups, the pair consisting of the group Spin(Π) and the set of canonical embeddings τi:GiSpin(Π), τij:GijSpin(Π) 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 GiSpin(2) are isomorphic to one another, it in fact suffices to fix one group USpin(2) instead with connecting homomorphisms ϕiji:UGij.

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 Gi are only given up to isomorphism, these standard embeddings are only well-defined up to automorphism of Gi, which leads to some ambiguity. Since by [21] the group K(Π) (and therefore each of its central extensions by a finite group) is a topological group, one may assume the ϕiji to be continuous, thus restricting oneself to the ambiguity stemming from the two continuous automorphisms of Spin(2), the identity and the inversion homomorphisms. This ambiguity is resolved in Section 10.

Theorem A provides us with the means of characterizing Wspin(Π).

Let Π be an irreducible simply-laced Dynkin diagram, I={1,,n} a set of labels of the vertices of Π, and for each iI let

  • τi:GiSpin(2)Spin(Π) be the canonical enveloping homomorphisms,

  • xiGi be elements of order eight whose polar coordinates involve the angle π4,

  • ri:=τi(xi).

Then Wspin(Π):=riiI satisfies the defining relations

ri8=1,(R1)ri-1rj2ri=rj2ri2n(i,j)for ijI,(R2)rirjrimij factors=rjrirjmij factorsfor ijI,(R3)

where

mij={3,if i, j form an edge,2,if i, j do not form an edge,

and

n(i,j)={1,if i, j form an edge,0,if i, j do not form an edge.

To be a set of defining relations means that any product of the ri that in Wspin(Π) 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 Spin(Π) and K(Π) 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 Spin(Π) 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 Wspin 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 SO(n) and Spin(n) as universal enveloping groups of SO(2)-, resp. Spin(2)-amalgams of type An-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 SO(2)- and Spin(2)-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 Spin(Π) and K(Π) onto non-trivial compact Lie groups.

I Basics

2 Conventions

:={1,2,3,} denotes the set of positive integers.

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

For any group G, consider the following maps:

inv:GG,xx-1,the inverse map,sq:GG,xx2,the square map.

Both maps commute with any group homomorphism.

For any group G, we denote by Z(G) 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.𝒞], [25, Section 1.3], [14, Section 1].

Let U be a group, and I a set. A U-amalgam over I is a set

𝒜={Gij,ϕijiijI}

such that Gij is a group and ϕiji:UGij is a monomorphism for all ijI. The maps ϕiji are called connecting homomorphisms. The amalgam is continuous if U and Gij are topological groups, and ϕiji is continuous for all ijI.

Let 𝒜~={G~ij,ϕ~ijiijI} and 𝒜={Gij,ϕijiijI} be U-amalgams over I. An epimorphism, resp. an isomorphism α:𝒜~𝒜 of U-amalgams is a system

α={π,αijijI}

consisting of a permutation πSym(I) and group epimorphisms, resp. isomorphisms

αij:G~ijGπ(i)π(j)

such that for all ijI

αijϕ~iji=ϕπ(i)π(j)π(i),

that is, the following diagram commutes:

More generally, let

𝒜~={G~ij,ϕ~ijiijI}

be a U-amalgam and let

𝒜={Gij,ϕijiijI}

be a V-amalgam. An epimorphism α:𝒜~𝒜 is a system

α={π,ρi,αijijI}

consisting of a permutation πSym(I), group epimorphisms ρi:UV and group epimorphisms

αij:G~ijGπ(i)π(j)

such that for all ijI,

αijϕ~iji=ϕπ(i)π(j)π(i)ρπ(i),

that is, the following diagram commutes:

If (and only if) in the epimorphism α:𝒜~𝒜 each ρi:UV is an isomorphism, then one obtains an epimorphism α:𝒜~𝒜 of U-amalgams by defining α={π,αijijI} and 𝒜={Gij,(ϕiji)} via

(ϕiji):UGij,u(ϕijiρi)(u).

If this α 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 Gi and a collection of groups Gij with connecting homomorphisms

ψiji:GiGij.

Since in our situation for all i there exist isomorphisms γi:UGi, it suffices to consider the connecting homomorphisms ϕiji=ψijiγi.

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

αi:GiG¯π(i)andαij:GijG¯π(i)π(j)

such that

αijψiji=ψ¯π(i)π(j)π(i)αi.

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 𝒜={Gij,ϕijiijI}, an enveloping group of 𝒜 is a pair (G,τ) consisting of a group G and a set

τ={τijijI}

of enveloping homomorphisms τij:GijG such that

G=τij(Gij)ijI

and

τijϕijj=τkjϕkjjfor all ijkI,

that is, for ijkI the following diagram commutes:

We write τ:𝒜G and call τ an enveloping morphism. An enveloping group (G,τ) and the corresponding enveloping morphism are faithful if τij is a monomorphism for all ijI.

Given a U-amalgam 𝒜={Gij,ϕiji}, an enveloping group (G,τ) is called a universal enveloping group if, given an enveloping group (H,τ) of 𝒜, there is a unique epimorphism ψ:GH such that for all i,jI with ij one has ψτij=τij. We write τ:𝒜G and call τ a universal enveloping morphism. By universality, two universal enveloping groups (G1,τ1) and (G2,τ2) of a U-amalgam 𝒜 are (uniquely) isomorphic.

The canonical universal enveloping group of the U-amalgam 𝒜 is the pair (G(𝒜),τ^), where G(𝒜) is the group given by the presentation

G(𝒜):=ijIGij|all relations in Gij, and ϕijj(x)=ϕkjj(x)for all ijkI and all xU

and where τ^={τ^ijijI} with the canonical homomorphism τ^ij:GijG(𝒜) for all ijI. 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

  • 𝒜~={G~ij,ϕ~ijiijI} is a U -amalgam over I,

  • 𝒜={Gij,ϕijiijI} is a V -amalgam over I,

  • α={π,ρi,αijijI} is an amalgam epimorphism 𝒜~𝒜,

  • (G,τ) with τ={τijijI}) is an enveloping group of 𝒜.

Then the following hold:

  • (a)

    There is a unique enveloping group (G,τ~), τ~={τ~ijijI} , of 𝒜~ such that the following diagram commutes for all ijI :

  • (b)

    Suppose (G~,τ~), τ~={τ~ijijI} , is a universal enveloping group of 𝒜~ . Then there is a unique epimorphism α^:G~G such that the following diagram commutes for all ijI :

  • (c)

    If α is an isomorphism and (G,τ) is a universal enveloping group, then α^ is also an isomorphism.

Proof.

(a) Let ijI. Since αij is an epimorphism, we must have

τ~ij:=τπ(i)π(j)αij

for the diagrams to commute; the claimed uniqueness follows. The fact that αij is an epimorphism also implies

τ~ij(G~ij)=(τπ(i)π(j)αij)(G~ij)=τπ(i)π(j)(Gπ(i)π(j)),

and so

G=τij(G~ij)=τ~ij(G~ij).

Moreover, for ijkI we find

τ~ijϕ~ijj=τπ(i)π(j)αijϕ~ijj=τπ(i)π(j)ϕπ(i)π(j)π(j)ρπ(j)=τπ(k)π(j)ϕπ(k)π(j)π(j)ρπ(j)=τπ(k)π(j)αkjϕ~kjj=τ~kjϕ~kjj.

Hence (G,{τ~ij}) is indeed an enveloping group of 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 α^ making the upper right triangle commute. The claim follows.

(c) This follows from part (b) by interchanging the roles of Gπ(i)π(j), G and G~ij, 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=SO(2), U~=Spin(2), V~={±1}.

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

  • 𝒜~={G~ij,ϕ~ijiijI} is a U~ -amalgam over I such that

    G~ij=ϕ~iji(U~),ϕ~ijj(U~),

  • 𝒜={Gij,ϕijiijI} is a U -amalgam over I,

  • α={π,ρi,αijijI} is an amalgam epimorphism 𝒜~𝒜,

  • (G~,τ~) with τ~={τ~ijijI}) is a universal enveloping group of 𝒜~,

  • (G,τ) with τ={τijijI}) is a universal enveloping group of 𝒜,

  • α^:G~G is the epimorphism induced by α via the commutative diagrams ( ijI )

    as in Lemma 7 (b).

For i,jI, with ij define Ziji:=ϕ~iji(V~) and Zijϕ~:=Ziji,Zijj, as well as Aij:=ker(αij). Then if

AijZijϕ~Z(G~ij),

it follows that G~ is a central extension of G by N:=τ~ij(Aij)ijI.

In this situation the epimorphism α:A~A is called an |N|-fold central extension of amalgams.

Proof.

We proceed by proving the following two assertions:

  • (a)

    NZ(G~),

  • (b)

    G~/NG.

Consider the following commutative diagram:

For ijI set

Zi:=τ~ij(Ziji)=(τ~ijϕ~iji)(V~)andG~i:=(τ~ijϕ~iji)(U~).

The hypothesis implies

τ~ij(G~ij)=G~i,G~jandG~=τ~ij(G~ij)ijI=G~iiI.

Moreover,

Zi=τ~ij(Ziji)τ~ij(Z(G~ij))Z(τ~ij(G~ij))=Z(G~i,G~j),

whence Zi centralises G~j for all i,jI. Since G~ is generated by the G~i, one has

ZiiIZ(G~iiI)=Z(G~).

Therefore

N=τ~ij(Aij)ijIτ~ij(Ziji)ijI=ZiiIZ(G~),

i.e., (a) holds.

Commutativity of the diagram implies Nker(α^) and so the homomorphism theorem yields an epimorphism G~/NG, gNα^(g). We construct an inverse map by exploiting that G and G~ are universal enveloping groups of 𝒜, resp. 𝒜~, in order to show that this epimorphism actually is an isomorphism. Indeed, for gGij, let g~αij-1(g). Then

τ~ij(αij-1(g))=τ~ij(g~Aij)τ~ij(g~)N=τ~ij(αij-1(g))NG~/N.

Thus one obtains a well-defined homomorphism

τ^π(i)π(j):Gπ(i)π(j)G~/N,gτ~ij(αij-1(g))N.

Then (G~/N,{τ^ij}) is an enveloping group for 𝒜. In particular, for uU and ijkI one has

(τ^π(i)π(j)ϕπ(i)π(j)π(j))(u)=τ~ij(αij-1(ϕπ(i)π(j)π(j)(u)))=τ~ij(ϕ~ijj((ρj)-1(u)))=τ~kj(ϕ~kjj((ρj)-1(u)))=τ~kj(αkj-1(ϕπ(k)π(j)π(j)(u)))=(τ^π(k)π(j)ϕπ(k)π(j)π(j))(u).

Since (G,{τij}) is a universal enveloping group of 𝒜, there exists a unique epimorphism β:GG~/N such that for ijI one has

βτij=τ^ij.

By the definition of α^ and τ^ij one finds

α^τ^ij=τij.

Therefore

(βα^)τ^ij=τ^ijand(α^β)τij=τij.

But (G,τ) and (G~,τ~) are universal enveloping groups; their uniqueness property implies that βα^=idG~/N and α^β=idG and hence as claimed G~/NG. 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=(a(i,j))i,jI such that for all ijI,

  • (a)

    a(i,i)=2,

  • (b)

    a(i,j) is a non-positive integer,

  • (c)

    if a(i,j)=0 then a(j,i)=0.

The matrix A is of two-spherical type if a(i,j)a(j,i){0,1,2,3} for all i,jI with ij.

A Dynkin diagram (or short: diagram) is a graph Π with vertex set V(Π) and edge set E(Π)(V(Π)2) such that each edge has an edge valency of 1, 2, 3 or and, in addition, edges with valency 2 or 3 are directed. If {v,w}E(Π) is directed from v to w, we write vw. Let E0(Π):=(V(Π)2)E(Π), and let E1(Π), E2(Π), E3(Π), resp. E(Π) be the subsets of E(Π) of edges of valency 1, 2, 3, resp. . The elements of E1(Π), E2(Π), E3(Π) are called edges of type A2, C2 resp. G2. 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 . If V(Π) is finite, then a labelling of Π is a bijection σ:IV(Π), where I:={1,,|V(Π)|}.

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

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

Conversely, given a two-spherical Dynkin diagram Π with vertex set V, we obtain a two-spherical generalised Cartan matrix A(Π):=(a(i,j))i,jI over I:=V by setting for ijI,

a(i,i):=2,a(i,j):={0,if {i,j}E(Π),-2,if {i,j}E2(Π) and ij,-3,if {i,j}E3(Π) and ij,-1,otherwise.

These two operations are inverse to each other, i.e., we have Π(A(Π))=Π and A(Π(A))=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 Π(A) to it by labelling the edge between i and j with whenever a(i,j)a(j,i)4. In this case it is, of course, not possible to reconstruct the values of a(i,j) and a(j,i) from the diagram Π.

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

Figure 1

An augmented Dynkin diagram.

5 The groups SO(n) and O(n)

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

Given a quadratic space (𝕂,V,q) with dim𝕂V<, we set

O(q):={aGL(V)q(av)=q(v) for all vV},SO(q):=O(q)SL(V).

Given n, let qn:n, xi=1nxi2 be the standard quadratic form on n, and

O(n):={aGLn()aat=En}O(qn)=O(-qn),SO(n):=O(n)SLn()SO(qn)=SO(-qn)O(qn)=O(-qn).

Since an element of O(n) has determinant 1 or -1, we have [O(n):SO(n)]=2.

Let n and let =(e1,,en) be the standard basis of n. Given a subset I{1,,n}, we set

I:={eiiI},VI:=In,qI:=qn|VI:VI.

There are canonical isomorphisms

M:End(n)Mn(),aM(a)

and

MI:End(VI)M|I|(),aMI(a)

that map an endomorphism into its transformation matrix with respect to the standard basis , resp. the basis I. Moreover, there is a canonical embedding

εI:O(qI)O(qn),

inducing a canonical embedding

MεIMI-1:O(|I|)O(n),

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

εij:SO(2)SO(n).

6 The groups Spin(n) and Pin(n)

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

Let (,V,q) be a quadratic space and let T(V)=n0Vn be the tensor algebra of V. The identity V0= provides a ring monomorphism T(V), and the identity V1=V a vector space monomorphism VT(V); these allow one to identify , V with their respective images in T(V). For

(q):=vv-q(v)vV

define the Clifford algebra of q as

Cl(q):=T(V)/(q).

Moreover, let

Cl(q)*:={xCl(q)there exists yCl(q) such that xy=1}.

The transposition map is the involution

τ:Cl(q)Cl(q)induced byv1vkvkv1,viV,

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

Π:Cl(q)Cl(q)given byv1vk(-1)kv1vk,viV,

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

Cl0(q):={xCl(q)Π(x)=x}

and

Cl1(q):={xCl(q)Π(x)=-x},

which yields a 2-grading of Cl(q), i.e.,

Cl(q)=Cl0(q)Cl1(q)

and

Cli(q)Clj(q)Cli+j(q) for i,j2.

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

σ:Cl(q)Cl(q),xx¯:=τΠ(x)=Πτ(x),

and the spinor norm

N:Cl(q)Cl(q),xxx¯.

In the following, (,V,q) is an anisotropic quadratic space such that dimV<.

Given xCl(q)*, the map

ρx:Cl(q)Cl(q),yΠ(x)yx-1

is the twisted conjugation with respect to x . Using the canonical identification of V with its image in Cl(q), we define

Γ(q):={xCl(q)*ρx(v)V for all vV}

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

ρ:Γ(q)GL(V),xρx,

which is the twisted adjoint representation.

Given n and V=n, we set

Cl(n):=Cl(-qn)andΓ(n):=Γ(-qn).

Recall that qn is defined to be the standard quadratic form on n (cf. Definition 1) and note that in the literature one can also find the opposite sign convention.

(a) Let n and let e1,,en be the standard basis of n. Then the following hold in Cl(n) for 1ijn,

ei2=-1,eiej=-ejei,(eiej)2=-1.

The first identity is immediate from the definition. The second identity follows from polarization, as in the tensor algebra T(n) one has

(qn)(ei+ej)(ei+ej)-q(ei+ej)=eiei+eiej+ejei+ejej-q(ei)-q(ej)-2b(ei,ej)=eiej+ejei,

where b(,) denotes the bilinear form associated to qn. The third identity is immediate from the first two.

(b) One has Cl0(3), where denotes the quaternions. Indeed, given a basis e1, e2, e3 of 3, a basis of Cl0(3), considered as an -vector space, is given by 1, e1e2, e2e3, e3e1. 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:Cl(q)Cl(q) induces a homomorphism

N:Γ(q)*

such that

N(Π(x))=N(x)for all xΓ(q).

Proof.

Cf. [13, Proposition 1.9]. ∎

The group

Pin(q):={xΓ(q)N(x)=1}Γ(q)

is the pin group with respect to q , and

Spin(q):=Pin(q)Cl0(q)Pin(q)

is the spin group with respect to q . By Lemma 6 and the 2-grading of Cl(q), the sets Pin(q) and Spin(q) are indeed subgroups of Γ(q). Given n, define

Pin(n):=Pin(-qn)andSpin(n):=Spin(-qn).

The following hold:

  • (a)

    One has [Pin(q):Spin(q)]=2 and Spin(q)=ρ-1(SO(q)).

  • (b)

    The twisted adjoint representation ρ:Γ(q)GL(V) induces an epimorphism ρ:Pin(q)O(q) . In particular, given n , we obtain epimorphisms

    ρn:=Mρ:Pin(n)O(n),ρn:=Mρ:Spin(n)SO(n)

    with ker(ρn)={±1} in both cases.

  • (c)

    The group Spin(q) is a double cover of the group SO(q).

Proof.

See [13, Theorem 1.11]. ∎

(a) By slight abuse of notation, suppressing the choice of basis, we will also sometimes denote the map ρn by ρ.

(b) Let H1Spin(n) and H2Pin(n) be such that -1H1 and -1H2, respectively, and let H~i:=ρn(Hi). Then we have Hi=ρn-1(H~i). We will explicitly determine these groups for some canonical subgroups of SO(n) and O(n).

Let nN, let I{1,,n}, and let ε~I:Cl(-qI)Cl(-qn) be the natural embedding of algebras afforded by the inclusion

{eiiI}{ei1in}

of bases of VI, resp. V. Then ε~I restricts and corestricts to an embedding

Pin(-qI)Pin(-qn)

of groups such that the following diagram commutes:

In analogy to Notation 2 we will use the same symbol for the (co)restriction of ε~I to Spin(). The most important application of this map in this article is for |I|=2 with I={i,j} providing the map

ε~ij:Spin(2)Spin(n).

Proof.

Let xΓ(-qI). By definition,

ρε~I(x)(v)=(εIρ)(v)VIV=nfor all vVI.

Since eiej=-ejei for all ijI by Remark 5 (a), for each -basis vector y=ej1ejk of ε~I(Cl(-qI)) and all i{1,,n}\I one has

Π(y)ei=eiy.

Hence

Π(ε~I(x))ei=eiε~I(x)

and thus for all i{1,,n}\I,

ρε~I(x)(ei)=Π(ε~I(x))eiε~I(x)-1=ein.

As ε~I(Cl(-qI)) is generated as an -algebra by the set {eiiI}, we in particular have

ρε~I=εIρ.

Therefore εI(x)Γ(-qn). Finally,

N(ε~I(x))=ε~I(N(x))=ε~I(1)=1,

whence

ε~I(x)Pin(-qn).

Since ε~I(Spin(-qI))=eiejijICl0(-qn), one has

ε~I(Spin(-qI))Pin(-qn)Cl0(-qn)=Spin(-qn).

Let nN and I{1,,n}. Then

ρn-1(εI(O(|I|)))=ε~I(Pin(-qI))𝑎𝑛𝑑ρn-1(εI(SO(|I|)))=ε~I(Spin(-qI)).

Proof.

By Lemma 10, one has

ρnε~I(Pin(-qI))=εI(O(|I|))andρnε~I(Spin(-qI))=εI(SO(|I|)),

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

Let n, let I{1,,n} and let m:=|I|. Then there exists an isomorphism i:Pin(m)Pin(-qI) such that the following diagram commutes:

As in Notation 2 we slightly abuse notation and also write ε~I for the map

idε~Ii:Pin(m)Pin(n)

and εI for the map

MεεIMεI-1:O(m)O(n).

Consequently, we obtain the following commutative diagram:

According to [13, Corollary 1.12], the group Pin(n) is generated by the set {vnN(v)=1} and each element of the group Spin(n) can be written as a product of an even number of elements from this set. That is, each element gSpin(2) is of the form

g=i=12k(aie1+bie2)=i=1k((a2i-1a2i+b2i-1b2i)+(a2i-1b2i-a2ib2i-1)e1e2)=:a+be1e2.

The requirement aie1+bie2{vnN(v)=1} is equivalent to

ai2+bi2=(aie1+bie2)(-aie1-bie2)=(aie1+bie2)(aie1+bie2)¯=N(aie1+bie2)=1.

Moreover,

1=N(g)=N(a+be1e2)=(a+be1e2)(a+be1e2)¯=(a+be1e2)(a+be2e1)=a2+b2.

Certainly, Spin(2) contains all elements of the form a+be1e2 with a2+b2=1, i.e., one obtains

Spin(2)={cos(α)+sin(α)e1e2α}.

One has

(cos(α)+sin(α)e1e2)-1=cos(α)-sin(α)e1e2=cos(-α)+sin(-α)e1e2,

i.e., the map

Spin(2),αcos(α)+sin(α)e1e2

is a group homomorphism from the real numbers onto the circle group. The twisted adjoint representation ρ2 maps the element cos(α)+sin(α)e1e2Spin(2) to the transformation

x1e1+x2e2(cos(α)+sin(α)e1e2)(x1e1+x2e2)(cos(α)-sin(α)e1e2)=x1(cos(α)2-sin(α)2)e1-2x2cos(α)sin(α)e1+2x1cos(α)sin(α)e2+x2(cos(α)2-sin(α)2)e2=(x1cos(2α)-x2sin(2α))e1+(x1sin(2α)+x2cos(2α))e2,

i.e., the rotation of the Euclidean plane 2 by the angle 2α, corresponding to the matrix

(cos(2α)-sin(2α)sin(2α)cos(2α))SO(2).

In other words, ρ2 is the double cover of the circle group by itself, cf. Theorem 8 (b).

Similarly, each element gSpin(3) is of the form

g=i=12k(aie1+bie2+cie3)=a+be1e2+ce2e3+de3e1

and each element hSpin(4) of the form

h=i=12k(aie1+bie2+cie3+die4)=h1+h2e1e2+h3e2e3+h4e3e1+h5e1e2e3e4+h6e4e3+h7e4e1+h8e4e2.

7 The isomorphism Spin(4)Spin(3)×Spin(3)

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

Denote by

:={a+bi+cj+dka,b,c,d}

the real quaternions, identify with the centre of via , aa, let

¯:,x=a+bi+cj+dkx¯=a-bi-cj-dk

be the standard involution, and let

U1():={xxx¯=1}

be the group of unit quaternions.

By [13, Section 1.4] one has

Spin(3)U1()andSpin(4)Spin(3)×Spin(3)U1()×U1().

The isomorphism Spin(3)U1() in fact is an immediate consequence of the isomorphism Cl0(3) from Remark 5 (b) plus the observation that this isomorphism transforms the spinor norm into the norm of the quaternions.

A canonical isomorphism Spin(4)Spin(3)×Spin(3)U1()×U1() can be described as follows (see [13, Section 1.4]). By Remark 14 each element of Spin(4) is of the form

a+be1e2+ce2e3+de3e1+ae1e2e3e4+be4e3+ce4e1+de4e2.

For

i:=e1e2,j:=e2e3,k:=e3e1,𝕀:=e1e2e3e4,i:=e4e3,j:=e4e1,k:=e4e2

one has

ij=k,jk=i,ki=j,i𝕀=𝕀i=i,j𝕀=𝕀j=j,k𝕀=𝕀k=k,i2=j2=k2=-1,𝕀2=1,σ(𝕀)=𝕀,

where σ(𝕀) denotes the Clifford conjugate of 𝕀, cf. Definition 1. We conclude that for every xSpin(4) there exist uniquely determined u=a+bi+cj+dk, v=a+bi+cj+dk such that

x=u+𝕀v.

One computes

N(x)=N(u+𝕀v)=(u+𝕀v)(u¯+𝕀v¯)=uu¯+vv¯+𝕀(uv¯+vu¯),

i.e.,

N(x)=1uu¯+vv¯=1 and uv¯+vu¯=0.

Hence, for 1=N(x)=N(u+𝕀v), one has

N(u+v)=(u+v)(u¯+v¯)=1,N(u-v)=(u-v)(u¯-v¯)=1.

That is, the map

Spin(4)Spin(3)×Spin(3),u+𝕀v(u+v,u-v)

is a well-defined bijection and, since

(u+𝕀v)(u+𝕀v)=uu+vv+𝕀(uv+vu)

and

(u+v,u-v)(u+v,u-v)=(uu+vv+uv+vu,uu+vv-(uv+vu)),

it is in fact an isomorphism of groups.

Consequently, there exists a group epimorphism

η~:Spin(4)Spin(3),u+𝕀vu+v.

Using this isomorphism

Spin(4)Spin(3)×Spin(3)U1()×U1()

there exists a natural homomorphism

Spin(4)SO()SO(4),(a,b)(xaxb-1).

Note that the restrictions (a,1)(xax) and (1,b)(xxb-1) are both injections of Spin(3)U1() into GL()(\{0},), in fact into SO(), as the norm is multiplicative. Since the kernel of this action has order two, the homomorphism Spin(4)SO()SO(4) must be onto by Proposition 8. We conclude that the group SO(4) is isomorphic to the group consisting of the maps

,xaxb-1for a,bU1();

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

Spin(3)SO(i,j,k)SO(3),a(xaxa-1)

is an epimorphism and, thus, that the group SO(3) is isomorphic to the group consisting of the maps

,xaxa-1for aU1();

see also [37, Lemma 11.24].

There also exists a group epimorphism

η:SO(4)SO(3)

induced by the map

SO(4){,xaxb-1a,bU1(){,xaxa-1aU1()}SO(3)(xaxb-1)(xaxa-1).

Altogether, one obtains the following commutative diagram:

8 Lifting automorphism from SO(n) to Spin(n)

For SO(2)×SO(2)={(a,b)a,bSO(2)} let

ι1:SO(2)SO(2)×SO(2),x(x,1),

and

ι2:SO(2)SO(2)×SO(2),x(1,x).

Similarly, for Spin(2)×Spin(2)={(a,b)a,bSpin(2)} let

ι~1:Spin(2)Spin(2)×Spin(2),x(x,1),

and

ι~2:Spin(2)Spin(2)×Spin(2),x(1,x).

Moreover, define

ρ2×ρ2:Spin(2)×Spin(2)SO(2)×SO(2),(a,b)(ρ2(a),ρ2(b)).

Hence

(ρ2×ρ2)ι~1=ι1ρ2,(ρ2×ρ2)ι~2=ι2ρ2.

Furthermore, let

π:Spin(2)×Spin(2)Spin(2)×Spin(2)/(-1,-1)

be the canonical projection. By the homomorphism theorem of groups the map ρ2×ρ2 factors through Spin(2)×Spin(2)/(-1,-1) and induces the following commutative diagram:

For α let

D(α):=(cos(α)-sin(α)sin(α)cos(α))SO(2)

and

S(α):=cos(α)+sin(α)e1e2Spin(2).

Then Spin(2)={S(α)α} and SO(2)={D(α)α} and there is a continuous group isomorphism

ψ:SO(2)Spin(2),D(α)S(α).

By the computation in Remark 14 the epimorphism ρ2 from Theorem 8 satisfies ρ2=sqψ-1, i.e.

ρ2:Spin(2)SO(2),S(α)D(2α).

Given an automorphism γAut(SO(2)), there is a unique automorphism γ~Aut(Spin(2)) such that ρ2γ~=γρ2. Moreover, γ is continuous if and only if γ~ is continuous.

Proof.

Define γ~:=ψγψ-1. Then

ρ2γ~=(sqψ-1)(ψγψ-1)=sqγψ-1=γsqψ-1=γρ2.

Uniqueness follows as Aut(SO(2))Aut(Spin(2)), γψγψ-1 is an isomorphism. ∎

Given an automorphism γAut(SO(2)×SO(2)), there is a unique automorphism γ~Aut(Spin(2)×Spin(2)) such that

(ρ2×ρ2)γ~=γ(ρ2×ρ2).

Proof.

Let γ~:=ψγψ-1, where

ψ:SO(2)×SO(2)Spin(2)×Spin(2),(D(α),D(β))(S(α),S(β)),

and observe that ρ2×ρ2=sqψ-1. The claim now follows as in the proof of Proposition 2. ∎

Let n3. Given an automorphism γAut(SO(n)), there is a unique automorphism γ~Aut(Spin(n)) such that

ρnγ~=γρn.

Proof.

For n3, both SO(n) and Spin(n) are perfect, cf. [24, Corollary 6.56]. By Theorem 8 (b) the group Spin(n) is a central extension of SO(n). Since Spin(n) is simply connected (see, e.g., [13, Section 1.8], it is in fact the universal central extension of SO(n).

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

γ~,γ~:Spin(n)Spin(n)

such that

γρn=ρnγ~andγ-1ρn=ρnγ~.

Hence

ρnγ~γ~=γρnγ~=γγ-1ρn=ρn

and, similarly,

ρnγ~γ~=ρn.

The universal property therefore implies γ~γ~=id=γ~γ~, i.e., γ~ 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 εij in Notation 2 and of ε~ij in Lemma 10.

Let ϕ:Spin(2)Spin(n) be a homomorphism such that

ker(ρnϕ)={1,-1}𝑎𝑛𝑑ρnϕ=εijρ2

for some ijI. Then ϕ=ε~ij.

Proof.

By Consequence 12 one has

ϕ(Spin(2))(ρn-1εijρ2)(Spin(2))=ρn-1(εij(SO(2)))=ε~ij(Spin(2)).

By hypothesis, kerϕ{1,-1}. If -1kerϕ, then

1=ϕ(-1)=ϕ(S(π))=ϕ(S(π2))2,

i.e., we have ϕ(S(π2)){1,-1}, whence S(π2)ker(ρnϕ), a contradiction. Consequently, ϕ is a monomorphism.

Consider the following commuting diagram:

One has ρ2ε~ij-1ϕ=ρ2=idρ2. Since ϕ is injective, the map ε~ij-1ϕ is an automorphism of Spin(2). Hence Proposition 2 implies ε~ij-1ϕ=id. ∎

II Simply-laced diagrams

9 SO(2)-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 SO(2) instead of the perfect group SU(2). This leads to some subtle complications that we will need to address below.

Recall the maps ε12,ε23:SO(2)SO(3) from Notation 2 and the maps ι1,ι2:SO(2)SO(2)×SO(2) from Notation 1.

Let Π be a simply-laced diagram with labelling σ:IV. An SO(2)-amalgam with respect to Π and σ is an amalgam

𝒜={Gij,ϕijiijI}

such that

Gij={SO(3),if {i,j}σE(Π),SO(2)×SO(2),if {i,j}σE(Π),for all ijI

and for i<jI,

ϕiji(SO(2))={ε12(SO(2)),if {i,j}σE(Π),ι1(SO(2)),if {i,j}σE(Π),ϕijj(SO(2))={ε23(SO(2)),if {i,j}σE(Π),ι2(SO(2)),if {i,j}σE(Π).

The standard SO(2)-amalgam with respect to Π and σ is the SO(2)-amalgam

𝒜(Π,σ,SO(2)):={Gij,ϕijiijI}

with respect to Π and σ with

ϕiji={ε12,if {i,j}σE(Π),ι1,if {i,j}σE(Π),ϕijj={ε23,if {i,j}σE(Π),ι2,if {i,j}σE(Π),

for all i<jI.

The key difference between the standard SO(2)-amalgam and an arbitrary SO(2)-amalgam 𝒜={Gij,ϕijiijI} with respect to Π and σ is that, for instance, ε12-1ϕiji can be an arbitrary automorphism of SO(2). 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 SO(2) whereas it does hold for the group SO(3). Hence, obviously, not every automorphism of SO(2) is induced by an automorphism of SO(3) and so it is generally not possible to undo the automorphism ε12-1ϕiji inside SO(3). 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 SO(2)-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 SO(2)-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 SO(2), SO(2)×SO(2), SO(3), 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:=(1-11)SO(3).

Then the map γD:SO(3)SO(3), ADAD-1=DAD is an automorphism of SO(3) such that

γDε12=ε23𝑎𝑛𝑑γDε23=ε12.

Proof.

Given (xy-yx)SO(2), we have

(1-11)(xy-yx1)(1-11)=(1y-xxy)(1-11)=(1xy-yx).

The second assertion follows analogously. ∎

The only influence of the labelling σ of an amalgam is the choice of which of the vertices iσ, jσ corresponds to which subgroup of Gij. We now show that this choice does not affect the isomorphism type of the amalgam.

Let Π be a simply-laced diagram with labellings σ1,σ2:IV. Then

𝒜(Π,σ1,SO(2))𝒜(Π,σ2,SO(2)).

Proof.

Denote 𝒜:=𝒜(Π,σ1,SO(2)) and 𝒜¯:=𝒜(Π,σ2,SO(2)). Let DSO(3) be as in Lemma 4 and let π:=σ2-1σ1Sym(I). Notice that

G¯π(i)π(j)=SO(3){π(i),π(j)}σ2E(Π){i,j}πσ2E(Π){i,j}σ1σ2-1σ2E(Π){i,j}σ1E(Π)Gij=SO(3).

Given i<jI with {i,j}σ1E(Π), let

αij:={idSO(3),if π(i)<π(j),γD,if π(i)>π(j),

and given i<jI with {i,j}σ1E(Π), let

αij:SO(2)×SO(2)SO(2)×SO(2),(x,y){(x,y),if π(i)<π(j),(y,x),if π(i)>π(j).

Then the system α:={π,αijijI}:𝒜𝒜¯ is an isomorphism of amalgams. Indeed, given i<jI with {i,j}σ1E(Π), one has

αijϕiji=αijε12={idSO(3)ε12=ε12=ϕ¯π(i)π(j)π(i),if π(i)<π(j),γDε12=ε23=ϕ¯π(i)π(j)π(j),if π(i)>π(j),

and

αijϕijj=αijε23={idSO(3)ε23=ε23=ϕ¯π(i)π(j)π(i),if π(i)<π(j),γDε23=ε12=ϕ¯π(i)π(j)π(j),if π(i)>π(j).

The case i<jI with {i,j}σ1E(Π) is verified similarly. ∎

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

Let B:=(1-I2), C:=(-I21)SO(3). Then the following holds:

  • (a)

    The map γB:SO(3)SO(3), ABAB-1 is an automorphism of SO(3) such that

    γBε12=ε12inv𝑎𝑛𝑑γBε23=ε23.

  • (b)

    The map γC:SO(3)SO(3), ACAC-1 is an automorphism of SO(3) such that

    γCε12=ε12𝑎𝑛𝑑γCε23=ε23inv.

Proof.

(a) Given A:=(xy-yx)SO(2), we have

γB(ε12(A))=B(xy-yx1)B-1=(x-yyx1)=ε12(A-1).

The second statement follows analogously.

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

Let Π be a simply-laced diagram with labelling σ:IV and let A={Gij,ϕijiijI} be a continuous SO(2)-amalgam with respect to Π and σ. Then AA(Π,σ,SO(2)).

Proof.

Denote 𝒜¯:=𝒜(Π,σ,SO(2)). The only continuous automorphisms of the circle group SO(2) are id and the inversion inv. Since 𝒜 is continuous by hypothesis, for all i<jI with {i,j}σE(Π) we have

ϕiji{ε12,ε12inv},ϕijj{ε23,ε23inv}.

Let B,CSO(3) be as in Lemma 7, let π:=idI, and given i<jI such that {i,j}σE(Π), let

αij:={idSO(3),if ϕiji=ε12,ϕijj=ε23,γB,if ϕiji=ε12inv,ϕijj=ε23,γC,if ϕiji=ε12,ϕijj=ε23inv,γBγC,if ϕiji=ε12inv,ϕijj=ε23inv.

For i<jI with {i,j}σE(Π), define

αij:=(ι1(ϕiji)-1)×(ι2(ϕijj)-1).

Then the system α:={π,αijijI}:𝒜𝒜¯ is an isomorphism of amalgams. ∎

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

For n3, the group SO(n) is a universal enveloping group of the amalgam A(An-1,SO(2)).

Proof.

Let I:={1,,n-1}. The group SO(n) acts flag-transitively on the simply connected projective geometry 𝒢:=n-1(); 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 SLn(). A maximal flag is given by

e1e1,e2e1,,en-1.

Let T be the subgroup of SO(n) of diagonal matrices; it is isomorphic to C2n-1 (where C2 is a cyclic group of order 2). For 1in-1, let HiSO(2) be the circle group acting naturally on ei,ei+1 and, for 1in-2, let Hi,i+1SO(3) be the group acting naturally on ei,ei+1,ei+2. Furthermore, for all i and j with 1i<j-1n-2, let Hij:=HiHjSO(2)×SO(2). Then the stabiliser of a sub-flag of co-rank one is of the form HiT, 1in-1, and the stabiliser of a sub-flag of co-rank two is of the form HijT, 1i<jn-1.

The group TC2n-1 admits a presentation with all generators and relations contained in the rank two subgroups Hij of SO(n): Indeed, T is generated by the groups C2Ti:=-1HiSO(2) and for each 1ijn-1 the relation TiTj=TjTi is visible within Hij. 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:=SO(n) is the universal enveloping group of the amalgam 𝒜(𝒢,H)={Hij,ΦijiijI}, where Φiji:HiHij is the inclusion map for each ijI. One has

Hi=ε{i,i+1}(SO(2)),iI,

and for all i<jI,

Hij={ε{i,i+1,i+2}(SO(3)),if j=i+1,ε{i,i+1}(SO(2))×ε{j,j+1}(SO(2)),if ji+1.

As a consequence, the system

α={idI,αij,αiijI}:𝒜(An-1,SO(2))𝒜(𝒢,H)

with

αi=ε{i,i+1}:SO(2)Hi,iI,

and for all i<jI,

αij={ε{i,i+1,i+2},if j=i+1,ε{i,i+1}×ε{j,j+1},if ji+1,

is an isomorphism of amalgams. ∎

The above proof mainly relies on geometric arguments in the Tits building of type An-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 Spin(2)-amalgams of simply-laced type

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

Recall the maps ε~12,ε~23:Spin(2)Spin(3) from Lemma 10 and the maps ι~1,ι~2:Spin(2)Spin(2)×Spin(2) from Notation 1.

Let Π be a simply-laced diagram with labelling σ:IV. A Spin(2)-amalgam with respect to Π and σ is an amalgam

𝒜={Gij,ϕijiijI}

such that for all ijI,

Gij={Spin(3),if {i,j}σE(Π),Spin(2)×Spin(2),if {i,j}σE(Π),

and for i<jI,

ϕiji(Spin(2))={ε~12(Spin(2)),if {i,j}σE(Π),ι1(Spin(2)),if {i,j}σE(Π),ϕijj(Spin(2))={ε~23(Spin(2)),if {i,j}σE(Π),ι2(Spin(2)),if {i,j}σE(Π).

The standard Spin(2)-amalgam with respect to Π and σ is the (continuous) Spin(2)-amalgam

𝒜(Π,σ,Spin(2)):={Gij,ϕijiijI}

with respect to Π and σ with

ϕiji={ε~12,if {i,j}σE(Π),ι~1,if {i,j}σE(Π),ϕijj={ε~23,if {i,j}σE(Π),ι~2,if {i,j}σE(Π).

for i<jI.

Proposition 2 enables us to lift SO(2)-amalgams to Spin(2)-amalgams: Let Π be a simply-laced diagram with labelling σ:IV and let

𝒜={Gij,ϕijiijI}

be an SO(2)-amalgam with respect to Π and σ. Given i,jI with i<j, there are γiji,γijjAut(SO(2)) such that

ϕiji={ε12γiji,if {i,j}σE(Π),ι1γiji,if {i,j}σE(Π),ϕijj={ε23γijj,if {i,j}σE(Π),ι2γijj,if {i,j}σE(Π).

We then lift γiji,γijj as in Lemma 2 to γ~iji,γ~ijjAut(Spin(2)) and set

ϕ~iji:={ε~12γ~iji,if {i,j}σE(Π),ι~1γ~iji,if {i,j}σE(Π),ϕ~ijj:={ε~23γ~ijj,if {i,j}σE(Π),ι~2γ~ijj,if {i,j}σE(Π),

and

G~ij:={Spin(3),if {i,j}σE(Π),Spin(2)×Spin(2),if {i,j}σE(Π).

Let Π be a simply-laced diagram with labelling σ:IV and let 𝒜={Gij,ϕijiijI} be an SO(2)-amalgam with respect to Π and σ. Then

𝒜~:={G~ij,ϕ~ijiijI}

is the induced Spin(2)-amalgam with respect to Π and σ. We also set

ρij:={ρ3,if {i,j}σE(Π),ρ2×ρ2,if {i,j}σE(Π).

Let Π be a simply-laced diagram with labelling σ:IV, let A={Gij,ϕijiijI} be an SO(2)-amalgam with respect to Π and σ, and let ϕ~iji be as introduced in Notation 2. Then for all ijI

ϕijiρ2=ρijϕ~iji.

Proof.

Without loss of generality suppose i<jI. Let γiji,γijjAut(SO(2)), γ~iji,γ~ijjAut(Spin(2)), and let ϕ~iji and ϕ~ijj be as introduced in Notation 2. Then, if {i,j}σE(Π), we find that

ϕijiρ2=ε12γijiρ2=ε12ρ2γ~iji=ρ3ε12γ~iji(by Remark 6.13)=ρ3ϕ~iji.

Similarly we also conclude

ϕijjρ2=ρ3ϕ~ijj.

Moreover, in case {i,j}σE(Π) we deduce

ϕijiρ2=ι1γijiρ2=ι1ρ2γ~iji=(ρ2×ρ2)ι~1γ~iji=(ρ2×ρ2)ϕ~iji.

Again we conclude by a similar argument that also

ϕijjρ2=(ρ2×ρ2)ϕ~ijj.

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

πΠ,σ={idI,ρ2,ρij}:𝒜(Π,σ,Spin(2))𝒜(Π,σ,SO(2)).

Let Π be a simply-laced diagram with labelling σ:IV, let A1 and A2 be SO(2)-amalgams with respect to Π and σ, and let

α={π,αijijI}:𝒜1𝒜2

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

α~={π,α~ijijI}:𝒜~1𝒜~2

such that for all ijI,

ρπ(i)π(j)α~ij=αijρij.

Proof.

Suppose 𝒜1={Gij,ϕijiijI} and 𝒜2={Hij,ψijiijI}. Let ijI. Since both amalgams are defined with respect to Π and σ, and since GijHπ(i)π(j), we conclude that {i,j}σ is an edge if and only if {i,j}πσ is an edge. Hence up to relabelling and identifying Gij with its image under αij, we may assume π=id and Gij=Hij and, thus, αijAut(Gij) with αijϕiji=ψiji. We distinguish two cases.

Case I: {𝐢,𝐣}𝛔𝐄(Π). Then G~ij=Spin(3). Let α~ijAut(Spin(3)) be the unique automorphism from Proposition 4 satisfying ρ3α~ij=αijρ3. It remains to verify that this is compatible with the amalgam structure. Indeed,

ρ3α~ijϕ~iji(ψ~iji)-1=αijρ3ϕ~iji(ψ~iji)-1=αijϕijiρ2(ψ~iji)-1(by Lemma 10.4)=αijϕiji(ψiji)-1=idε12(SO(2))ρ3(by Lemma 10.4)=ρ3idε~12(Spin(2)).

Hence, by uniqueness in Proposition 4, one has

α~ijϕ~iji(ψ~iji)-1=idε~12(Spin(2)),

i.e., α~ijϕ~iji=ψ~iji.

Case II: {𝐢,𝐣}𝛔𝐄(Π). In this case, we have G~ij=Spin(2)×Spin(2). Let α~ijAut(Spin(2)×Spin(2)) be the unique automorphism from Corollary 3 satisfying (ρ2×ρ2)α~ij=αij(ρ2×ρ2). It remains to verify that this is compatible with the amalgam structure. Indeed,

(ρ2×ρ2)α~ijϕ~iji(ψ~iji)-1=αij(ρ2×ρ2)ϕ~iji(ψ~iji)-1=αijϕijiρ2(ψ~iji)-1(by Lemma 10.4)=αijϕiji(ψiji)-1=idι1(SO(2))(ρ2×ρ2)(by Lemma 10.4)=(ρ2×ρ2)idι~1(Spin(2)).

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

α~ijϕ~iji=ψ~iji.

Let Π be a simply-laced diagram with labellings σ1,σ2. Then A(Π,σ1,Spin(2))A(Π,σ2,Spin(2)).

Proof.

Let 𝒜1:=𝒜(Π,σ1,SO(2)) and 𝒜2:=𝒜(Π,σ2,SO(2)). The definitions imply 𝒜~1=𝒜(Π,σ1,Spin(2)) and 𝒜~2=𝒜(Π,σ2,Spin(2)). Moreover, one has 𝒜1𝒜2 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 𝒜(Π,Spin(2)) to denote the isomorphism type of 𝒜(Π,σ,Spin(2)) and, by slight abuse of notation, any representative of this isomorphism type. It is called the standard Spin(2)-amalgam with respect to Π.

Let Π be a simply-laced diagram with labelling σ:IV and let A~ be a continuous Spin(2)-amalgam with respect to Π and σ. Then

𝒜~𝒜(Π,Spin(2)).

Proof.

Let 𝒜 be the continuous SO(2)-amalgam that induces 𝒜~, which exists by Remark 5. By Theorem 8, one has 𝒜𝒜(Π,SO(2)). Proposition 6 yields the claim, since 𝒜(Π,SO(2)) induces 𝒜(Π,Spin(2)). ∎

For n3, the group Spin(n) is the universal enveloping group of A(An-1,Spin(2)).

Proof.

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

e1e1,e2e1,,en-1.

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:=Spin(n) is the universal enveloping group of the amalgam

𝒜(𝒢,H)={Hij,ϕijiijI},

where the Hij are the “block-diagonal” rank two subgroups and Φiji:HiHij is the inclusion map for each ijI. By Consequence 12 and Remark 13, one has

Hi=ε~{i,i+1}(Spin(2)),iI,

and for all i<jI,

Hij={ε~{i,i+1,i+2}(Spin(3)),if j=i+1,ε~{i,i+1}(Spin(2))ε~{j,j+1}(Spin(2)),if ji+1.

As a consequence, the system

α={idI,αij,αiijI}:𝒜(An-1,Spin(2))𝒜(𝒢,H)

with

αi=ε~{i,i+1}:Spin(2)Hi,iI,

and for all i<jI,

αij={ε~{i,i+1,i+2},if j=i+1,ε~{i,i+1}ε~{j,j+1},if ji+1,

is an epimorphism of Spin(2)-amalgams. In fact, each αi and each αii+1 is an isomorphism, only the αij:Spin(2)×Spin(2)Spin(2).Spin(2), ji+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 Spin(n) is an enveloping group of the amalgam 𝒜(An-1,Spin(2)). It remains to prove universality.

Let (G,τij) be an arbitrary enveloping group of 𝒜(An-1,Spin(2)) and let i and j be given with 1i<jn-1 with ji+1. By definition the following diagram commutes for 1abcn-1:

In particular, one has

(τi,i+2ϕi,i+2i)(-1)=(τi,i+1ϕi,i+1i)(-1)(set b=i, a=i+1, c=i+2)=(τi,i+1ϕi,i+1i+1)(-1)(since ε~12(-1)=ε~23(-1))=(τi+1,i+2ϕi+1,i+2i+1)(-1)(set b=i+1, a=i, c=i+2)=(τi+1,i+2ϕi+1,i+2i+2)(-1)(since ε~12(-1)=ε~23(-1))=(τi,i+2ϕi,i+2i+2)(-1)(set b=i+2, a=i, c=i+1).

We conclude by induction that τij:Spin(2)×Spin(2)G, ji+1, always factors through Spin(2).Spin(2) or, in other words, τ:𝒜(An-1,Spin(2))G always factors through 𝒜(𝒢,H). That is, the universal enveloping group Spin(n) of 𝒜(𝒢,H) is also a universal enveloping group of 𝒜(An-1,Spin(2)). ∎

The proof of Theorem 10 would become a bit easier if one replaced Spin(2)×Spin(2) by Spin(2).Spin(2) 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 Π=(V,E) be a (finite) simply-laced diagram with labelling σ:IV and let c(Π) denote the number of connected components of Π. A component labelling of Π is a map 𝒦:V{1,,c(Π)} such that u,vV are in the same connected component of Π if and only if 𝒦(u)=𝒦(v).

Throughout this section, let Π be a (finite) simply-laced diagram, σ:IV a labelling and 𝒦 a component labelling.

Generalizing Theorem 9, the universal enveloping group of a continuous SO(2)-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 Spin(2)-amalgam over the same simply-laced diagram Π. In the case of E10 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 SO(2)- and Spin(2)-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(Π) be the corresponding simply connected split real Kac–Moody group, and let K(Π) be its maximal compact subgroup, i.e., the subgroup fixed by the Cartan–Chevalley involution. Then there exists a faithful universal enveloping morphism

τK(Π):𝒜(Π,SO(2))K(Π).

Proof.

For iI denote by Gi the fundamental rank one subgroups of G(Π) and, for ijI, by Gij the fundamental rank two subgroups. The groups Gi are isomorphic to SL2() and the groups Gij are isomorphic to SL3() or to SL2()×SL2(), depending on whether the vertices iσ, jσ of Π are joined by an edge or not. The Cartan–Chevalley involution ω leaves the groups Gi, Gij setwise invariant and, in fact, induces the transpose-inverse map on these groups. Define Hi:=FixGi(ω)SO(2) and Hij:=FixGij(ω), the latter being isomorphic to SO(3) or to SO(2)×SO(2), and let ψiji:HiHij denote the (continuous) inclusion map for ijI. By [12, Theorem 1.2], the group K(Π) is the universal enveloping group of the amalgam 𝒜1:={Hij,ψijiijI}.

Given i<jI such that {i,j}σE(Π), by Theorem 8 applied to the subdiagram of Π of type A2 consisting of the vertices iσ, jσ there is a continuous isomorphism αij:HijSO(3) such that

(αijψiji)(Hi)=ε12(SO(2)),(αijψijj)(Hj)=ε23(SO(2)).

Let iI and choose jI such that {i,j}σE(Π). Define

αi:={ε12-1αijψiji:HiSO(2),if i<j,ε23-1αijψiji:HiSO(2),if i>j.

For i<jI such that {i,j}σE(Π), define

αij:=αi×αj:Hij=Hi×HjSO(2)×SO(2).

For arbitrary ijI, let

Kij:={SO(3),if {i,j}σE(Π),SO(2)×SO(2),if {i,j}σE(Π),

and

ϕ¯iji:=αijψijiαi-1:SO(2)Kij.

Then 𝒜2:={Kij,ϕ¯ijiijI} is an SO(2)-amalgam with respect to Π and σ. Moreover, the system α={idI,αij,αiijI}:𝒜1𝒜2 is an isomorphism of amalgams. Indeed, given ijI, one has

ϕ¯ijiαi=αijψijiαi-1αi=αijψiji.

Finally, αi is continuous for each iI, whence ϕ¯iji is continuous for all ijI. Therefore, 𝒜2 is a continuous SO(2)-amalgam with respect to Π and σ so that 𝒜1𝒜2𝒜(Π,SO(2)) by Theorem 8. ∎

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

𝒜(Π,SO(2))={Kij,ϕijiijI}.

Similarly for the standard Spin(2)-amalgam with respect to Π (cf. Definition 8):

𝒜(Π,Spin(2))={K~ij,ϕ~ijiijI}.

We denote the epimorphism of amalgams from Remark 5 by

πΠ:𝒜(Π,Spin(2))𝒜(Π,SO(2)).

As discussed in Remark 2 the amalgam 𝒜(Π,SO(2)) consists of compact Lie groups with continuous connecting homomorphisms. On the other hand, the group K(Π) naturally carries a Hausdorff group topology that is kω: Indeed, K(Π) 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(Π) studied in [21, Section 7] fixed by the Cartan–Chevalley involution; both ambient groups are kω (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 kω-group topologies on K(Π) induced from the real Kac–Moody group G(Π) and from the unitary form coincide, as both are induced from the kω-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 kω-group topology is the finest group topology with respect to the enveloping homomorphisms τij:KijK(Π). In other words, the obvious analog of [14, Theorem 6.12] and [21, Theorem 7.22] holds for (K(Π),τK(Π)). In particular, to any enveloping morphism ψ=(ψij):𝒜(Π,SO(2))X into a Hausdorff topological group X with continuous homomorphisms ψij:KijX there exists a unique continuous homomorphism ξ:K(Π)X such that the following diagram commutes:

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

The spin group Spin(Π) with respect to Π is the canonical universal enveloping group of the (continuous) amalgam 𝒜(Π,Spin(2)) with the canonical universal enveloping morphism

τSpin(Π)={τijijI}:𝒜(Π,Spin(2))Spin(Π).

K(Π) is an enveloping group of the amalgam A(Π,Spin(2)). There exists a unique central extension ρΠ:Spin(Π)K(Π) that makes the following diagram commute (cf. Notation 8):

where

τK(Π)={ψij:KijK(Π)ijI}:𝒜(Π,SO(2))K(Π)

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

Proof.

As in Definition 3, for ijI let ρij:K~ijKij be the epimorphism ρ3 if {i,j}σ is an edge, and ρ2×ρ2 otherwise. Then, by Lemma 7(a), the group K(Π) with the homomorphisms

ξij:=ψijρij:K~ijK(Π)

for all ijI is an enveloping group of 𝒜(Π,σ,Spin(2)). By universality of τSpin(Π):𝒜(Π,Spin(2))Spin(Π), Lemma 7(b) provides a unique epimorphism ρΠ:Spin(Π)K(Π) 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 ijI and kI. If iσ and kσ are in the same connected component of Π, then

τij(ϕ~iji(-1Spin(2)))=τk(ϕ~kk(-1Spin(2))).

Proof.

As iσ and kσ are in the same connected component, there exists a sequence i0:=i,i1,,in:=kI such that {irσ,ir+1σ} are edges for 0r<n. Thus K~irir+1=Spin(3) and by the definition of Spin(Π) as the canonical universal enveloping group of the amalgam 𝒜(Π,Spin(2)), we have

ϕ~irir+1ir(-1Spin(2))=-1Spin(3)=ϕ~irir+1ir+1(-1Spin(2)).

Hence

τij(ϕ~iji(-1Spin(2)))=τi0i1(ϕ~i0i1i0(-1Spin(2)))=τi0i1(ϕ~i0i1i1(-1Spin(2)))=τin-1in(ϕ~in-1inin(-1Spin(2)))=τin-1k(ϕ~in-1kk(-1Spin(2)))=τk(ϕ~kk(-1Spin(2))).

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

Thus the following is well-defined.

For ijI define

-1Spin(Π),𝒦(i):=τij(ϕ~iji(-1Spin(2)))

and

Z:=-1Spin(Π),1,,-1Spin(Π),c(Π)Spin(Π).

The following are true:

  • (a)

    Z is contained in the centre of Spin(Π).

  • (b)

    |Z|2c(Π).

Assertion (a) is immediate from Proposition 9 applied to the Spin(2)-amalgam 𝒜(Π,Spin(2)) and the SO(2)-amalgam 𝒜(Π,SO(2)) with U~=Spin(2), V~=-1 and U=SO(2). The second follows from the fact that Z is abelian by assertion (a) and admits a generating system of c(Π) involutions by definition.

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

One has |Z|=2c(Π).

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

Let V be an R-vector space and let Xi,XjEnd(V) be such that

Xi2=-idV=Xj2,XiXj=-XjXi.

Then the map

ψ:Spin(3)GL(V),a+be1e2+ce2e3+de1e3aidV+bXi+cXj+dXiXj

is a group monomorphism such that

ψ(ε~12(S(α)))=cos(α)idV+sin(α)Xi,ψ(ε~23(S(α)))=cos(α)idV+sin(α)Xj.

Proof.

The subspace :=idV,Xi,Xj,XiXj is an -subalgebra of End(V), the set {idV,Xi,Xj,XiXj} is an -basis of , and the -linear extension

ψ^:Cl(3)0

of

1idV,e1e2Xi,e1e3XiXj,e2e3Xj

is an isomorphism of algebras: Indeed, since idV, Xi, Xj and XiXj satisfy the same relations as 1, e1e2, e2e3 and e1e3, the map ψ^ is a homomorphism of rings. Since Xi0End(V), one has ker(ψ^)Cl(3)0. By Remark 5, Cl(3)0 is a skew field and, thus, simple as a ring. Therefore, ψ^ is injective and, hence, bijective, because dim4, i.e., ψ^ is an isomorphism of algebras.

Consequently, the restriction ψ of ψ^ to Spin(3) is injective with values in the group GL(V), i.e., ψ:Spin(3)GL(V) is a group monomorphism. The final statement is immediate from the definitions. ∎

Let V be an R-vector space and let Xi,XjEnd(V) be such that

XjidV,Xi,Xi2=-idV=Xj2,XiXj=XjXi.

Then the map

ψ:ε~12(Spin(2)),ε~34(Spin(2))Spin(4)GL(V),a+be1e2+ce3e4+de1e2e3e4aidV+bXi+cXj+dXiXj

is a group monomorphism such that

ψ(ε~12(S(α)))=cos(α)idV+sin(α)Xi,ψ(ε~34(S(α)))=cos(α)idV+sin(α)Xj.

Proof.

The subspace 𝔸:=idV,Xi,Xj,XiXj is an -subalgebra of End(V), the set {idV,Xi,Xj,XiXj} is an -basis of 𝔸, and the -linear extension

ψ^:𝔸~:=1,e1e2,e3e4,e1e2e3e4Cl(4)0𝔸

of

1idV,e1e2Xi,e3e4Xj,e1e2e3e4XiXj

is an isomorphism of algebras: Indeed, since idV, Xi, Xj and XiXj satisfy the same relations as 1, e1e2, e3e4 and e1e2e3e4, the map ψ is a homomorphism of rings. The hypothesis XjidV,Xi implies that the set {1,Xi,Xj,XiXj} is -linearly independent. It follows that ψ^ is injective and, thus, bijective, because dim𝔸4.

Consequently, the restriction ψ of ψ^ to ε~12(Spin(2)),ε~34(Spin(2))Spin(4) is injective with values in GL(V), i.e.,

ψ:ε~12(Spin(2)),ε~34(Spin(2))Spin(4)GL(V)

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 Spin(Π) 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 σ:IV,

  • 𝔤 be the Kac–Moody algebra corresponding to Π and 𝔨 its maximal compact subalgebra with Berman generators Y1 , …, Yn (cf. [ 20 , Section 1.2] ),

  • μ:𝔨End(s), s , be a maximal generalised spin representation (cf. [ 20 , Corollary 3.10] ),

  • Xi:=2μ(Yi) for each iI.

Then, for each ijI, there exist subgroups XijGLs(C) and an enveloping morphism

Ψ𝒜={ψijijI}:𝒜(Π,Spin(2))X:=XijijI

with injective ψij whenever {i,j}σE(Π).

Proof.

According to [20, Remark 3.7], given ijI, one has

Xi2=-idV=Xj2,

and

XiXj={-XjXi,if {i,j}σE(Π),XjXi,if {i,j}σE(Π).

Moreover, XjidV,Xi, as μ is maximal. Thus Lemma 11 provides group monomorphisms

ψij:K~ijGLs(),

if {i,j}σE(Π), and Lemma 12 provides group homomorphisms

ψij:K~ijGLs()

with kernel (-1,-1), if {i,j}σE(Π). This allows one to define

Xij:=im(ψij).

Restriction of the ranges of the maps ψij to Xij thus provides group isomorphisms

ψij:K~ij=Spin(3)Xij,

if {i,j}σE(Π), and group epimorphisms

ψij:K~ij=Spin(2)×Spin(2)Xij,

if {i,j}σE(Π), satisfying for all ijI,

ψij(ϕ~ijj(cos(α)+sin(α)e1e2))=cos(α)idV+sin(α)Xj.

In particular, one has for all ijk,

ψijϕ~ijj=ψkjϕ~kjj.

The set Ψ𝒜:={ψijijI} is the desired enveloping morphism. ∎

Let everything be as in Theorem 14. By universality of

τSpin(Π):𝒜(Π,Spin(2))Spin(Π)

(cf. Definition 5) there exists an epimorphism

Ξ:Spin(Π)X

such that the following diagram commutes:

The commutative diagram in Lemma 6 and the finiteness of the central extension Spin(Π)K(Π) by Observation 9 in fact allow one to lift the topological universality statement from Remark 4 concerning

τK(Π):𝒜(Π,SO(2))K(Π)

to a topological universality statement concerning

τSpin(Π):𝒜(Π,Spin(2))Spin(Π).

Moreover, the maps ψ constructed in Lemmas 11 and 12 are certainly continuous with respect to the Lie group topologies, if dim(V)<.

In particular, the enveloping morphism Ψ𝒜={ψij} from the theorem consists of continuous maps, so that by universality Ξ:Spin(Π)X is continuous as well.

As an immediate consequence we record:

Let Π be an irreducible simply-laced diagram. Then one has 1Spin(Π)-1Spin(Π).

Proof.

We conclude from Remark 15

Ξ(-1Spin(Π))=(Ξτ12ϕ~121)(-1Spin(2))=(ψ12ϕ~121)(-1Spin(2))=cos(π)idV+sin(π)X1=-idV,

and, hence, 1Spin(Π)-1Spin(Π). ∎

Let Π be a simply-laced diagram. Then the universal enveloping group (Spin(Π),τSpin(Π)={τijijI}) of A(Π,Spin(2)) is a 2c(Π)-fold central extension of the universal enveloping group K(Π) of A(Π,SO(2)).

Proof.

Let Π1,,Πc(Π) be the connected components of Π. Then

Spin(Π)=Spin(Π1)××Spin(Πc(Π)).

Indeed,

τSpin(Π):𝒜(Π,Spin(2))Spin(Π1)××Spin(Πc(Π))

with

τij:K~ijτij(K~ij)

if 𝒦(i)=𝒦(j) and

τij:K~ij(τijϕ~iji)(Spin(2))×(τijϕ~ijj)(Spin(2)),(x,y)τij(x,y)

if 𝒦(i)𝒦(j) 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 U~-amalgam 𝒜(Π,Spin(2)) and the U-amalgam 𝒜(Π,SO(2)) with U~=Spin(2), U=SO(2) and V~=-1 combined with Lemma 7 and Corollary 16. ∎

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

III Non-simply-laced rank two diagrams

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, C2 and G2, we consider point-line models of the Tits buildings of the split real Lie groups Sp4() and G2(2), 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 U2()Sp4() and SO(4)G2(2) act flag-transitively on the respective point-line geometries.

Their unique double covers SO(2)×SU2()U2() and Spin(4)SO(4) fit into the commutative diagrams

and

which allow one to transform point and line stabilisers in U2() and SO(4) into point, resp. line stabilisers in SO(3) in a way that is compatible with the covering maps. This in turn will allow us to transform a Spin(2)-amalgam for a given two-spherical diagram Π into a Spin(2)-amalgam for the simply-laced diagram Πsl 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 Spin(Π) is a non-trivial central extension of K(Π) for suitable two-spherical diagrams Π.

As a caveat we point out that the compatibility of the covering maps in the C2 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

(2-r-s2)

for r,s such that rs4. 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

KK1TK*TKK2TK,

where

K1SO(2)K2

with TK={1,t1,t2,t1t2}/2×/2 and TKK1=t1, TKK2=t2 and KiKiTK. We conclude that the isomorphism type of K is known once the action of t1 on K2 and the action of t2 on K1 are known. It turns out that t1 centralises K2 if and only if r is even and inverts K2 if and only if r is odd; similarly, t2 centralises K1 if and only if s is even and inverts K1 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: KSO(2)×SO(2), if both r and s are even; KSO(3), if both r and s are odd; KU2(), 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 C2 and G2, enable us to replace edges labelled 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=(a(i,j))i,jI

be an irreducible doubly-laced diagram that admits two root lengths. Then the unfolded Dynkin diagram is the simply-laced Dynkin diagram Πun with type set

Iun:={±iiI,i short root}{iiI,i long root}

and edges defined via the generalised Cartan matrix Aun=(aun(i,j))i,jIun given by

aun(i,j)={0,if |i|, |j| have different lengths and a(|i|,|j|)=0,-1,if |i|, |j| have different lengths and a(|i|,|j|)0,a(|i|,|j|),if |i|, |j| have the same length and ij>0,0,if |i|, |j| have the same length and ij<0,

(cf. Definition 2).

There exists an embedding of K(Π) into K(Πun) that by Corollary 6 allows one to related the respective spin covers to one another.

13 Diagrams of type G2

In this section we prepare Strategy 1 for diagrams of type G2.

Denote by :={a+bi+cj+dka,b,c,d} the real quaternions. Then the standard involution of is given by

¯:,x=a+bi+cj+dkx¯=a-bi-cj-dk.

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

Pu:={xx=-x¯}={bi+cj+dkb,c,d}.

The split Cayley algebra 𝕆 is defined as the vector space endowed with the multiplication

xy=(x1,x2)(y1,y2)=(x1y1+y2x2¯,y1x2+x1¯y2),

cf. [7, Section 5.1]. The real split Cayley hexagon () consists of the one- and two-dimensional real subspaces of 𝕆 for which the restriction of the multiplication map is trivial, i.e., ()=(𝒫,,) with the point set

𝒫:={xx𝕆,x2=0x}

and the line set

:={x,yxy𝒫,xy=0},

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

Let x=(x1,x2)O. Then x2=0 if and only if x1PuH and x1¯x1-x2¯x2=0.

Proof.

Suppose x1Pu and x1¯x1-x2¯x2=0. Then

-x1x1-x2x2¯=x1¯x1-x2¯x2=0

and

x1x2+x1¯x2=x1x2-x1x2=0,

and, thus, x2=0. Conversely, suppose x2=0. Then

0=x2=(x1x1+x2x2¯,x1x2+x1¯x2),

so if x2=0, then x1=0, and there is nothing to show. For x20 multiplication of x1x2+x1¯x2=0 from the right with x2-1 gives x1+x1¯=0 or, equivalently, x1Pu. But now 0=x1x1+x2x2¯=-x1¯x1+x2¯x2, and the claim follows. ∎

Let N:, xxx¯ be the norm map associated to the standard involution of the real quaternions. Moreover, let

U1():={xxx¯=1}

be the group of real quaternions of norm one.

By Remark 3 (see also [37, Lemma 11.22]), the group SO(4) is isomorphic to the group consisting of the maps

,xaxb-1for a,bU1().

The group SO(4){HH,xaxb-1a,bU1(H)} acts flag-transitively on the split Cayley hexagon H(R) via

(x1,x2)(ax1a-1,ax2b-1).

Proof.

We show that the map

fa,b:𝕆𝕆:(x1,x2)(ax1a-1,ax2b-1)

is an algebra automorphism of 𝕆 for all a,bU1(). Then it also induces an automorphism of (), because it is defined via the multiplication in 𝕆. We have

fa,b(x1,x2)fa,b(y1,y2)=(ax1a-1,ax2b-1)(ay1a-1,ay2b-1)=((ax1a-1)(ay1a-1)+(ay2b-1)(ax2b-1)¯,(ay1a-1)(ax2b-1)+(ax1a-1)¯(ay2b-1))=(a(x1y1+y2x2¯)a-1,a(y1x2+x1¯y2)b-1)=fa,b((x1,x2)(y1,y2)),

whence the map fa,b is multiplicative. Since it is certainly an -linear bijection, it is an algebra automorphism of 𝕆. Flag-transitivity is an immediate consequence of the Iwasawa decomposition and the fact that SO(4) is the maximal compact subgroup of the (simply connected semisimple) split real group G2(2) of type G2. ∎

There exists a nice direct proof of flag-transitivity without making use of the Iwasawa decomposition and the structure theory of G2(2) 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 SO(4) for our amalgamation problem.

Let

Pu𝕆:=Pu

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

x1¯x1-x2¯x2=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 SO(3) is isomorphic to the group consisting of the maps

PuPu:xaxa-1 for aU1()

and acts transitively on the set {{x,-x}Puxx¯=1}. Moreover, for each a,x,zU1(), there exists a unique solution bU1() for the equation

z=axb-1.

Hence SO(4) acts transitively on the set

{{(x,y),(-x,-y)}Pu×xx¯=1=yy¯}.

But this implies point transitivity on the projective real quadric x1¯x1-x2¯x2=0 in Pu𝕆, which, in turn, implies point transitivity on () by Lemma 3.

Now choose one point of (), say (i,i). Then a point y=(y1,y2) is collinear to this point if and only if

(iy1-y2i,y1i-iy2)=0y1=-iy2i.

So the question of transitivity of the stabiliser of (i,i) in SO(4) on the line pencil of (i,i) in () is equivalent to the question of transitivity of the stabiliser of i in SO(3) on the line pencil of i in the projective plane PuPu{0}Pu𝕆.

But since the latter is transitive, so is the former, and hence SO(4) acts flag-transitively on () by means of the maps given in Lemma 5.

We denote the embeddings of the circle group into SO(4) as panel stabilisers (stabilisers of line pencils or point rows) of the split Cayley hexagon by ηp resp. ηl. Concretely, one has up to choosing an orientation of the circle group and up to choosing the split Cayley hexagon or its dual,

ηp:SO(2)SO(4),

given by

D(α)(I2D(α))=ε34(D(α)),

and

ηl:SO(2)SO(4),

given by

D(α)D~(α):=(cos(2α)-sin(2α)cos(α)sin(α)-sin(α)cos(α)sin(2α)cos(2α))=ε14(D(2α))ε23(D(-α)).

Let B:=diag(-1,1,1,-1), C:=diag(-1,-1,1,1)U2(C). Then the following hold:

  • (a)

    The map γB:SO(4)SO(4), ABAB-1=BAB is an automorphism of SO(4) such that

    γBηp=ηpinv𝑎𝑛𝑑γBηl=ηl.

  • (b)

    The map γC:SO(4)SO(4), ACAC-1=CAC is an automorphism of SO(4) such that

    γCηp=ηp𝑎𝑛𝑑γCηl=ηlinv.

Proof.

Straightforward. ∎

In the following, let

η~p:Spin(2)Spin(4),S(α)ε~34(S(α)),η~l:Spin(2)Spin(4),S(α)ε~14(S(2α))ε~23(S(-α)),

and recall from Theorem 8 (b) that for n2 the map ρn:Spin(n)SO(n) is the twisted adjoint representation.

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

The map η~l is a monomorphism.

Proof.

For S(α)kerη~l one has

η~l(S(α))=(cos(2α)+sin(2α)e1e4)(cos(α)-sin(α)e2e3)=1

and, thus, απ. As η~l(S(π))=-1 and η~l(S(2π))=1, one obtains

kerη~l={S(α)α2π}={1}.

One has

ρ4η~p=ηpρ2,ρ4η~l=ηlρ2.

Proof.

For α,

(ρ4η~p)(S(α))=(ρ4ε~34)(S(α))=(ε34ρ)(S(α))=(ηpρ)(S(α))

and

(ρ4η~l)(S(α))=ρ4(ε~14(S(2α))ε~23(S(-α)))=ε14(D(4α))ε23(D(-2α))=(ηlρ2)(S(α)).

Let V:=H and E:={1,i,j,k}. Then the following hold:

  • (a)

    For a,bU1() the maps

    a:,xax𝑎𝑛𝑑rb:,xxb-1

    preserve the norm

    N:,xxx¯.

    In particular, a,rbSO()SO(4).

  • (b)

    The map

    U1()×U1()SO(4),(a,b)arb

    is a group epimorphism with kernel {(1,1),(-1,-1)}.

Proof.

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

For ={1,i,j,k} define

La:=M(a)SO(4),Rb:=M(rb)SO(4).

We observe the following.

  • (a)

    The map U1()×U1()SO(4) from Lemma 7(b) equals the covering map ρ4, cf. Remark 4.

  • (b)

    Given x=a+bi+cj+dkU1(), one has x-1=a-bi-cj-dk, and a short computation shows

    Lx=(a-b-c-dba-dccda-bd-cba),Rx=(abcd-ba-dc-cda-b-d-cba)

    as -linear maps via left action. Lemma 7(b) implies that for all elements x,yU1() one has LxRy=RyLx and that up to scalar multiplication with -1 the matrices Lx and Ry are uniquely determined by their product.

  • (c)

    The action from Lemma 5 translates into

    ω:SO(4)×()(),(LaRb,(x,y))(LaRax,LaRby).

  • (d)

    For α, one has ηp(D(α))=LxRx with x=cos(α2)+sin(α2)i, i.e.,

    (I2D(α))=(D(α2)D(α2))(D(-α2)D(α2)).

  • (e)

    For α, we have ηl(D(α))=LxRy with x=cos(α2)+sin(α2)k and y=cos(3α2)-sin(3α2)k, i.e.,

    ηl(D(α))=(cos(α2)-sin(α2)D(α2)sin(α2)cos(α2))(cos(3α2)-sin(3α2)D(-3α2)sin(3α2)cos(3α2)).

The subgroup of right translations Rb is normal in SO(4), the resulting quotient is isomorphic to SO(3). This canonical projection induces a surjection from the split Cayley hexagon onto the real projective plane, given by the projection (x1,x2)x1. (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 η1:SO(4)SO(3) such that

    η1ηp=ε23,η1ηl=ε12.

  • (b)

    There is an epimorphism η2:SO(4)SO(3) such that

    η2ηp=ε12,η2ηl=ε23.

Proof.

By Remark 3, the map

ψ:SO(4)ε{2,3,4}(SO(3)),LaRbLaRa

is an epimorphism (see also [37, Corollaries 11.23–11.24]). By Remark 13(d),

ψηp=ηp=ε34,

and, by Remark (e),

(ψηl)(D(α))=(cos(α2)-sin(α2)D(α2)sin(α2)cos(α2))(cos(α2)sin(α2)D(α2)-sin(α2)cos(α2))=(1D(α)1)=ε23(D(α)).

Therefore, the map η1:=ε{2,3,4}-1ψ has the desired properties. The existence of the map η2 now follows from Lemma 4. ∎

The final result of this section allows us to carry out Strategy 1 for edges of type G2 in Theorem 11 below.

The following statements hold:

  • (a)

    There is an epimorphism η~1:Spin(4)Spin(3) such that

    η~1η~p=ε~23,η~1η~l=ε~12.

  • (b)

    There is an epimorphism η~2:Spin(4)Spin(3) such that

    η~2η~p=ε~12,η~2η~l=ε~23.

Proof.

The map η~1:Spin(4)Spin(3), u+𝕀vu+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

ε23ρ2=ηηpρ2=ηρ4η~p=ρ3η~1η~p

and

ker(ρ3η~1η~p)=ker(ε23ρ2)=kerρ2={±1Spin(2)}.

Therefore ε~23=η~1η~p by Proposition 5. In particular, also the upper triangle of the diagram commutes. The second claim concerning η~1 follows by an analogous argument The claims concerning η~2 are now immediate by Lemma 4 and Proposition 4

14 Diagrams of type C2

In this section we prepare Strategy 1 for diagrams of type C2.

Let Sp4() be the matrix group with respect to the -basis e1, ie1, e2, ie2 of 2 leaving the real alternating form

((x1,x2,x3,x4),(y1,y2,y3,y4))x1y2-x2y1+x3y4-x4y3

invariant.

The maximal compact subgroup of Sp4() is the group U2(), embedded as follows. Let e1, e2 be the standard basis of 2 and consider U2() as the isometry group of the scalar product ((v1,v2),(w1,w2))v1¯w1+v2¯w2. Defining

x1:=Re(v1),x2:=Im(v1),x3:=Re(v2),x4:=Im(v2),y1:=Re(w1),y2:=Im(w1),y3:=Re(w2),y4:=Im(w2),

we compute

v1¯w1+v2¯w2=x1y1+x2y2+x3y3+x4y4+i(x1y2-x2y1+x3y4-x4y3).

Since two complex numbers are equal if and only if real part and imaginary part coincide, the group U2() preserves the form x1y2-x2y1+x3y4-x4y3 and we have found an embedding in Sp4(), acting naturally on the -vector space 2 with -basis e1, ie1, e2, ie2.

As U2() also preserves the form x1y1+x2y2+x3y3+x4y4 with respect to the -basis e1, ie1, e2, ie2 of 2 it is at the same time also a subgroup of O(4), in fact of SO(4), since U2() 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 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 -basis e1, ie1, e2, ie2 and the -alternating form on 2 given in Remark 2, the flag e1e1,e2 is an incident point-line pair of the resulting symplectic quadrangle of Sp4(). The stabiliser in U2() of the point e1 is isomorphic to O1()×U1() where the first factor acts diagonally on e1ie1 and the second factor acts naturally on e2. The stabiliser of the line e1,e2 is isomorphic to O2() acting diagonally on e1,e2ie1,ie2.

Recall the definition of D(α) from Notation 1. Let

ζp:SO(2)U2()Sp4()SO(4),D(α)(11cos(α)sin(α)-sin(α)cos(α))

and let

ζl:SO(2)U2()Sp4()SO(4),D(α)D~(-α):=(cos(α)sin(α)cos(α)sin(α)-sin(α)cos(α)-sin(α)cos(α))

be the embeddings of the circle group arising as point-stabilizing resp. line-stabilizing rank one groups as above with respect to the -bases e1, e2 of 2 and e1, ie1, e2, ie2 of 2.

In the following, we shall identify ={x+iyx,y} with {(x-yyx)x,y}. This identification, in particular, embeds SO(2) into as the unit circle group. This induces an embedding GL2()GL4(), gg~. Note that for α, this embedding yields D(α)~=D~(α)U2().

Let B:=diag(-1,1,-1,1), C:=diag(-1,-1,1,1)U2(C). Then the following hold:

  • (a)

    The map γB:U2()U2():ABAB-1=BAB is an automorphism of U2() such that

    γBζp=ζpinv𝑎𝑛𝑑γBζl=ζl.

  • (b)

    The map γC:U2()U2():ACAC-1=CAC is an automorphism of U2() such that

    γCζp=ζp𝑎𝑛𝑑γCζl=ζlinv.

Proof.

Straightforward. ∎

Let αR. Then D~(α)SU2(C) and (D(-α)D(α))SU2(C).

Proof.

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

det(D(-α)D(α))=D(-α)D(α)=I2=1.

Returning to the embedding U2()SO(4) mentioned in Remark 2, the group

SU2()={(x1+iy1x2+iy2-x2+iy2x1-iy2)|x1,x2,y1,y2,x12+x22+y12+y22=1}U2()

acts -linearly on 2 with transformation matrices

(x1-y1x2-y2y1x1y2x2-x2-y2x1y1y2-x2-y1x1)

with respect to the basis e1, ie1, e2, ie2. Remark 13 implies that the map

SU2()SO(4)

given by

(x1+iy1x2+iy2-x2+iy2x1-iy2)Rx1-y1i+x2j-y2k=(x1-y1x2-y2y1x1y2x2-x2-y2x1y1y2-x2-y1x1)

injects SU2() into SO(4). The restriction of this map to SO(2)SU2() by setting y1=0=y2 provides the transformations D~ from Definition 3. The group

{(cos(α)+isin(α)00cos(α)+isin(α))}U1()SO(2)

acts with transformation matrices

(D(α)00D(α))=(cos(α)-sin(α)00sin(α)cos(α)0000cos(α)-sin(α)00sin(α)cos(α)),

i.e., Remark 13 implies that the map

U1()SO(4)

given by

(cos(α)+isin(α)00cos(α)+isin(α))Lcos(α)+isin(α)=(cos(α)-sin(α)00sin(α)cos(α)0000cos(α)-sin(α)00sin(α)cos(α))

injects U1() into SO(4). Altogether, using Remark 2, we obtain the following commutative diagram:

Our candidate for a spin cover of the group U2() therefore is its double cover

U1()×SU2()U2(),(z,A)zA.

Note that the fundamental group of U2() equals , as the determinant map det:U2()U1()SO(2) induces an isomorphism of fundamental groups; its simply connected universal cover is isomorphic to ×SU2(). The above double cover is unique up to isomorphism, because has a unique subgroup of index two (cf. [22, Theorem 1.38]).

In the following, let

ζ~p:Spin(2)SO(2)×SU2()Spin(4),S(α)(D(-α),(D(α)D(-α))),ζ~l:Spin(2)SO(2)×SU2()Spin(4),S(α)(id,D~(-α))

and let

ρ^:SO(2)×SU2()U2(),(z,A)(zz)A.

Recall the maps

ρ2:Spin(2)SO(2),S(α)D(2α),sq:GG,xx2,inv:GG,xx-1.

One has

ζ~p=ε~34,ζ~lsq=ε~23ε~14

Proof.

Using the identification Spin(4)Spin(3)×Spin(3) 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

ε~34(cos(α)+sin(α)e1e2)=cos(α)+sin(α)e3e4=cos(α)-𝕀isin(α)by Remark 7.2=(cos(α)-isin(α),cos(α)+isin(α))by Remark 7.2=(Lcos(α)-isin(α),Rcos(α)-isin(α))by Lemma 13.12, Remark 13.13=(D(-α),(D(α)D(-α)))by Remark 14.7=ζ~p(cos(α)+sin(α)e1e2)

and

ε~23(cos(α)+sin(α)e1e2)ε~14(cos(α)+sin(α)e1e2)=(cos(α)+sin(α)e2e3)(cos(α)+sin(α)e1e4)=(cos(α))2+jcos(α)sin(α)+𝕀((sin(α))2-jcos(α)sin(α))by Remark 7.2=(1,(cos(α)+jsin(α))2)by Remark 7.2=(id,R(cos(α)+jsin(α))2)by Lemma 13.12, Remark 13.13=(id,D~(-2α))by Remark 14.7=(ζ~lsq)(cos(α)+sin(α)e1e2),

as desired. ∎

The following observation is the analog of Lemma 11.

One has

ρ^ζ~p=ζpρ2,ρ^ζ~lsq=ρ^sqζ~l=ζlρ2.

Moreover,

(ρ^)-1(ζp)(SO(2))Spin(2),(ρ^)-1(ζl)(SO(2)){1,-1}×SO(2).

Proof.

For α,

(ρ^ζ~p)(S(α))=ρ^(D(-α),(D(α)D(-α)))=(1SO(2)D(-2α))=ζp(D(2α))=(ζpρ2)(S(α))

and

(ρ^sqζ~l)(S(α))=ρ^(1SO(2),D~(-2α))=D~(-2α)=ζl(D(2α))=(ζlρ2)(S(α)).

For the second claim observe that

(ρ^)-1(ζp)(SO(2))={(D(-α),(D(α)D(-α)))|α}Spin(2),(ρ^)-1(ζl)(SO(2))={(±id,D~(-α))α}{1,-1}×SO(2).

One has

ζp=ε34,ζl=ε23ε14

Proof.

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

Given an automorphism τAut(U2(C)), there is a unique τ~Aut(SO(2)×SU2(C)) such that

ρ^τ~=τρ^.

Proof.

Let τAut(U2()). Consider the characteristic subgroups

H1:=Z(U2())={zI2zU1()SO(2)},H2:=[U2(),U2()]=SU2()

of U2(). Note that H1 and H2 are also characteristic in H:=H1×H2, as H1=Z(H) and H2=[H,H]. Hence Aut(H)=Aut(H1)×Aut(H2).

Let τ1:=τ|H1Aut(H1) and τ2:=τ|H2Aut(H2). Then the automorphism τ~:=(τ1,τ2)Aut(H) satisfies

(ρ^τ~)(z,A)=ρ^(τ1(z),τ2(A))=τ1(z)τ2(A)=τ(z)τ(A)=τ(zA)=(τρ^)(z,A)

for all (z,A)SO(2)×SU2(). Let ψ=(ψ1,ψ2)Aut(SO(2)×SU2()) be such that ρ^ψ=τρ^. Given zSO(2) and ASU2(), one has

ψ1(z)=ψ1(z)ψ2(I2)=(ρ^ψ)(z,I2)=(τρ^)(z,I2)=(ρ^τ~)(z,I2)=τ1(z)τ2(I2)=τ1(z)

and

ψ2(A)=(ρ^ψ)(1SO(2),A)=(ρ^τ~)(1SO(2),A)=τ2(A).

The final result of this section allows us to carry out Strategy 1 for edges of type C2 in Theorem 11 below.

There is an epimorphism ζ~:SO(2)×SU2(C)Spin(3) such that

ζ~ζ~p=ε~12inv,ζ~ζ~l=ε~23.

Proof.

The map

ψ:Spin(3)SU2(),a+be1e2+ce2e3+de3e1Ra+bi+cj+dk=(abcd-ba-dc-cda-b-d-cba)

is a group isomorphism by Remarks 5 (b) and 13 (b) (see also [37, 11.26]), with the convention that the matrix group SU2() acts -linearly on 2 with respect to the -basis 1, i=i1, j, k=ij as in Definition 3 and Notation 4. Let

ζ~:SO(2)×SU2()Spin(3),(z,A)ψ-1(A).

For α, one has

(ζ~ζ~p)(S(α))=ζ~(D(-α),(D(α)D(-α)))=cos(α)-sin(α)e1e2=(ε~12inv)(S(α))

and

(ζ~ζ~l)(S(α))=ζ~(1SO(2),D~(-α))=cos(α)+sin(α)e2e3=ε~23(S(α)).

The following statements hold:

  • (a)

    There exist epimorphisms ζ1,ζ2,ζ3:U2()SO(3) such that

    ζ1ζp=ε12inv,ζ1ζl=ε23,ζ2ζp=ε12,ζ2ζl=ε23,ζ3ζp=ε23,ζ3ζl=ε12.

  • (b)

    There exist epimorphisms ζ~2,ζ~3:SO(2)×SU2()Spin(3) such that

    ζ~2ζ~p=ε~12,ζ~2ζ~l=ε~23,ζ~3ζ~p=ε~23,ζ~3ζ~l=ε~12.

Proof.

The kernel of the epimorphism SO(2)×SU2()U2(), (z,A)zA is equal to (D(π),D~(π)). One has ζ~(D(π),D~(π))=ψ-1(-I4)=-1Spin(3), whence (D(π),D~(π))ker(ρ3ζ~). Therefore, the claim concerning ζ1 follows from Proposition 13 and the homomorphism theorem of groups.

The claims about ζ2, ζ3 then follow from Lemma 7, resp. Lemma 4. A subsequent application of Proposition 4 yields the claims about ζ~2, ζ~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 be such that rs4 and consider the generalised Cartan matrix of rank two given by

A:=(a(i,j))i{1,2}=(2-r-s2)

and a simply connected Kac–Moody root datum

𝒟=({1,2},A,Λ,(ci)i{1,2},(hi)i{1,2}).

Let G:=G(A):=G(𝒟) be the corresponding (simply connected) real Kac–Moody group of rank two, let T0 be the fundamental torus of G with respect to the fundamental roots α1, α2, and let

K:=Kr,s:=K(A)

be the subgroup consisting of the elements fixed by the Cartan–Chevalley involution with respect to T0 of G, i.e., K is the maximal compact subgroup of G.

Let GiSL2() be the corresponding fundamental subgroups of rank one and define

Ki:=Kir,s:=GiKr,sSO(2)andT:=T0K.

By (KMG3) (see [43, p. 545] or, e.g., [31, p. 84]), the torus T0 is generated by μhi for i=1,2 and μ\{0} arbitrary. The action of the Cartan–Chevalley involution on the torus is given by μhi(μ-1)hi=μ-hi. Hence

T=T0K={id=1h1=1h2,t1:=(-1)h1,t2:=(-1)h2,t1t2=(-1)h1+h2}/2×/2.

The group K is isomorphic to a free amalgamated product

K1T*TK2T.

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-=ω(c+) 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 K1T and K2T. The claim follows from [38, Chapter I, Section 5]. ∎

Since KiKiT, this free amalgamated product is fully determined by the intersections KiT and the action of T on each Ki. Note that (KMG3) implies TK1={1,t1} and TK2={1,t2}. The action of T on each Ki can be extracted from the action of T0 on each Gi, which according to [43, (4), p. 549] (or also [31, (5.1), p. 86]) is given by

txi(λ)t-1=xi(t(ci)λ)

for tT0, λ and root group functor xi. According to [43, Section 2, p. 544] (or also [31, Definitions 5.1 and 5.5]) one computes for i,j{1,2} that

ti(cj)=(-1)hi(cj)=(-1)hi(cj)=(-1)a(i,j).

We conclude that ti acts trivially on Kj, if and only if the entry a(i,j) of the generalised Cartan matrix is even; conversely it acts non-trivially (and hence by inversion) if and only if a(i,j) is odd.

In symbols, for

n(i,j):={0,if a(i,j) is even,1,if a(i,j) is odd.

and kjKj one has

ti-1kjti=kj-2n(i,j)kj.(15.1)

Let

θ1:=θ1r,s:SO(2)K1r,s,θ2:=θ2r,s:SO(2)K2r,s,

be continuous isomorphisms.

Let r,s such that rs4, let A=(2-r-s2). Then set

Hr,s:={SO(2)×SO(2),if rs0(mod2),SO(3),if rs1(mod2),U2(), otherwise.

Also set

δ1:={ι1,if rs0(mod2),ε12,if rs1(mod2),ζl,if r0,s1(mod2),ζp,if r1,s0(mod2),δ2:={ι2,if rs0(mod2)(see Notation 8.1),ε23,if rs1(mod2)(see Notation 5.2),ζp,if r0,s1(mod2)(see Definition 14.3),ζl,if r1,s0(mod2).

Let r,sN such that rs4, let A=(2-r-s2), let G(A) be the corresponding simply connected real Kac–Moody group, and let Kr,s be its maximal compact subgroup. Then there exists a group epimorphism θ:Kr,sHr,s such that

θθ1=δ1𝑎𝑛𝑑θθ2=δ2.

Proof.

By Lemma 3 the group K is isomorphic to K1T*TK2T with T={1,t1,t2,t1t2}/2×/2 and T0K1={1,t1}, T0K2={1,t2}; in particular, K is generated by K1SO(2) and K2SO(2). As K is a free amalgamated product it therefore suffices to define θ on each of the Ki and to verify that the actions of the ti on the Kj are compatible with the actions of the images of the ti on the images of the Kj. Define θ via

θ|K1:K1δ1(SO(2)),x(δ1θ1-1)(x)

and

θ|K2:K2δ2(SO(2)),x(δ2θ2-1)(x).

Then this is compatible with the action of T. Indeed, using Remark 4, one observes:

  • (a)

    Since rs0(mod2), the elements ti centralise the groups Kj, which is compatible with the fact that SO(2)×SO(2) is an abelian group.

  • (b)

    Since rs1(mod2), the element t1 inverts the group K2 and the element t2 inverts the group K1, which is compatible with the situation in SO(3) by Lemma 7.

  • (c)

    Since r0(mod2), s1(mod2), the element t1 centralises K2 and the element t2 inverts the group K1. This is compatible with the following computations (cf. Definition 3): for all gK2,

    θ(t1gt1)=(-10-100-10-1)(11xy-yx)(-10-100-10-1)=(11xy-yx)=θ(g),

    and for all gK1,

    θ(t2gt2)=(11-100-1)(xyxy-yx-yx)(11-100-1)=(x-yx-yyxyx)=θ(g-1).

  • (d)

    This is dual to (c). ∎

Let r,s such that rs4, let A=(2-r-s2). Then set

H~r,s:={Spin(2)×Spin(2),if rs0(mod2),Spin(3),if rs1(mod2),SO(2)×SU2(),otherwise.

Furthermore, set

δ~1:={ι~1,if rs0(mod2),ε~12,if rs1(mod2),ζ~l,if r0,s1(mod2),ζ~p,if r1,s0(mod2),δ~2:={ι~2,if rs0(mod2)(see Notation 8.1),ε~23,if rs1(mod2)(see Lemma 6.10),ζ~p,if r0,s1(mod2)(see Notation 14.8),ζ~l,if r1,s0(mod2).

The central extension

ρ¯:H~r,sHr,s

satisfies

ρ¯={ρ2×ρ2,if rs0(mod2)(see Notation 8.1),ρ3,if rs1(mod2)(see Theorem 6.8),ρ^,otherwise (see Notation 14.8).

Let Kr,s=K(A)=K1T*TK2T be as in Lemma 3 and let t1K1T, t2K2T as in Remark 4.

Define

ui:=θ(ti)(see Proposition 15.7),U:=u1,u2/2×/2.

Furthermore, define

U~:=ρ¯-1(U),K~i:=K~ir,s:=ρ¯-1(θ(Ki)),

and the spin extension

K~:=K~r,s:=K~(A):=K~1U~*U~K~2U~

of

K=Kr,s=K(A),

let

ρ^^:K~1U~*U~K~2U~K1T*TK2T

be the epimorphism induced by ρ¯|K~1, ρ¯|K~2 and let

θ~1:Spin(2)K~1andθ~2:Spin(2)K~2

be continuous monomorphisms such that the following diagrams commute for i=1,2:

One has K~1Spin(2) unless r0,s1(mod2) and K~2Spin(2) unless r1,s0(mod2), in which case the respective group is isomorphic to {1,-1}×SO(2) (cf. Lemma 10). Hence θ~1 actually is a (continuous) isomorphism unless r0,s1(mod2), in which case it is a (continuous) isomorphism onto the unique connected subgroup of index two of K~1.

For i=1 the map on the left-hand side of the above commutative diagram is

Spin(2)SO(2),S(α)D(α)if r0,s1(mod2)(see Notation 8.1),Spin(2)SO(2),xρ2(x)otherwise.

The dual statement holds for θ~2.

In particular, for i=2 the map on the left-hand side of the above commutative diagram is

Spin(2)SO(2),S(α)D(α)if r1,s0(mod2)(see Notation 8.1),Spin(2)SO(2),xρ2(x)otherwise.

Define

t~1:={θ~1(S(π)),if r0,s1(mod2),θ~1(S(π2)),otherwise,t~2:={θ~2(S(π)),if r1,s0(mod2),θ~2(S(π2)),otherwise.

The following is true by construction:

Let r,sN such that rs4, let A=(2-r-s2), let G(A) be the corresponding simply connected real Kac–Moody group, and let Kr,s be its maximal compact subgroup, and let K~r,s be its spin extension. Then there exists a group epimorphism θ~:K~r,sH~r,s such that

θ~θ~1=δ~1 and θ~θ~2=δ~2.

Moreover, the following diagram commutes:

where the epimorphism on the left-hand side is one of ρ2 or S(α)D(α) as described in Remark 9.

Furthermore, for {i,j}={1,2} and k~jK~j one has

t~i-1k~jt~i=k~j-2n(i,j)k~j.(15.2)

Identity (15.2) follows from identity (15.1) in Remark 4

The Cartan matrix of type C2 over {1,2} with short root α1 and long root α2 (i.e., 21; see Remark 3) is

(2-2-12),

cf. [6, p. 44]. The group Hr,s from Proposition 10 is of type C2 with short root α1 and long root α2 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

Spin(2)SO(2),S(α)D(α)

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 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 K~=K~1U~*U~K~2U~ one has