David Ghatei , Max Horn , Ralf Köhl and Sebastian Weiß

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

Published online: August 9, 2016

Abstract

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

1 Introduction

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

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

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

Theorem A

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 i I let G i Spin ( 2 ) and for each i j I let

G i j { 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 , j I with i < j , let ϕ i j i : G i G i j be the standard embedding as “upper-left diagonal block” and let ϕ i j j : G j G i j 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 < j I G i j ) / , where is the equivalence relation determined by

ϕ i j i ( x ) ϕ i k i ( x )

for all i j , k I and x G i . Furthermore, there exists a canonical two-to-one central extension Spin ( Π ) K ( Π ) .

The system { G i , G i j , ϕ i j i } is called an amalgam of groups, the pair consisting of the group Spin ( Π ) and the set of canonical embeddings τ i : G i Spin ( Π ) , τ i j : G i j Spin ( 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 G i Spin ( 2 ) are isomorphic to one another, it in fact suffices to fix one group U Spin ( 2 ) instead with connecting homomorphisms ϕ i j i : U G i j .

The formalization of the concept of standard embedding as “upper-left/lower right diagonal block” can be found in Section 10. Note that, since the G i are only given up to isomorphism, these standard embeddings are only well-defined up to automorphism of G i , which leads to some ambiguity. Since by [21] the group K ( Π ) (and therefore each of its central extensions by a finite group) is a topological group, one may assume the ϕ i j i 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 W spin ( Π ) .

Theorem B

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

  • τ i : G i Spin ( 2 ) Spin ( Π ) be the canonical enveloping homomorphisms,

  • x i G i be elements of order eight whose polar coordinates involve the angle π 4 ,

  • r i := τ i ( x i ) .

Then W spin ( Π ) := r i i I satisfies the defining relations

(R1) r i 8 = 1 ,
(R2) r i - 1 r j 2 r i = r j 2 r i 2 n ( i , j ) for i j I ,
(R3) r i r j r i m i j factors = r j r i r j m i j factors for i j I ,

where

m i j = { 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 r i that in W spin ( 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 W spin under the generalised spin representation is finite, generalizing [9, Lemma 2, p. 49].

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

Sections 13, 14 and 15 provide us with the necessary tools for generalizing our findings to arbitrary diagrams; they deal with equivariant coverings of the real projective plane by the split Cayley hexagon and the symplectic quadrangle and with coverings of the real projective plane and the symplectic quadrangle by trees. In Section 16 we study 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

Notation 1

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

Notation 2

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

Notation 3

For any group G, consider the following maps:

inv : G G , x x - 1 , the inverse map ,
sq : G G , x x 2 , the square map .

Both maps commute with any group homomorphism.

Notation 4

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].

Definition 1

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

𝒜 = { G i j , ϕ i j i i j I }

such that G i j is a group and ϕ i j i : U G i j is a monomorphism for all i j I . The maps ϕ i j i are called connecting homomorphisms. The amalgam is continuous if U and G i j are topological groups, and ϕ i j i is continuous for all i j I .

Definition 2

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

α = { π , α i j i j I }

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

α i j : G ~ i j G π ( i ) π ( j )

such that for all i j I

α i j ϕ ~ i j i = ϕ π ( i ) π ( j ) π ( i ) ,

that is, the following diagram commutes:

More generally, let

𝒜 ~ = { G ~ i j , ϕ ~ i j i i j I }

be a U-amalgam and let

𝒜 = { G i j , ϕ i j i i j I }

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

α = { π , ρ i , α i j i j I }

consisting of a permutation π Sym ( I ) , group epimorphisms ρ i : U V and group epimorphisms

α i j : G ~ i j G π ( i ) π ( j )

such that for all i j I ,

α i j ϕ ~ i j i = ϕ π ( i ) π ( j ) π ( i ) ρ π ( i ) ,

that is, the following diagram commutes:

Notation 3

If (and only if) in the epimorphism α : 𝒜 ~ 𝒜 each ρ i : U V is an isomorphism, then one obtains an epimorphism α : 𝒜 ~ 𝒜 of U-amalgams by defining α = { π , α i j i j I } and 𝒜 = { G i j , ( ϕ i j i ) } via

( ϕ i j i ) : U G i j , u ( ϕ i j i ρ i ) ( 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.

Remark 4

More generally, an amalgam can be defined as a collection of groups G i and a collection of groups G i j with connecting homomorphisms

ψ i j i : G i G i j .

Since in our situation for all i there exist isomorphisms γ i : U G i , it suffices to consider the connecting homomorphisms ϕ i j i = ψ i j i γ i .

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

α i : G i G ¯ π ( i ) and α i j : G i j G ¯ π ( i ) π ( j )

such that

α i j ψ i j i = ψ ¯ π ( i ) π ( j ) π ( i ) α i .

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

Definition 5

Given a U-amalgam 𝒜 = { G i j , ϕ i j i i j I } , an enveloping group of 𝒜 is a pair ( G , τ ) consisting of a group G and a set

τ = { τ i j i j I }

of enveloping homomorphisms τ i j : G i j G such that

G = τ i j ( G i j ) i j I

and

τ i j ϕ i j j = τ k j ϕ k j j for all i j k I ,

that is, for i j k I the following diagram commutes:

We write τ : 𝒜 G and call τ an enveloping morphism. An enveloping group ( G , τ ) and the corresponding enveloping morphism are faithful if τ i j is a monomorphism for all i j I .

Definition 6

Given a U-amalgam 𝒜 = { G i j , ϕ i j i } , 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 : G H such that for all i , j I with i j one has ψ τ i j = τ i j . We write τ : 𝒜 G and call τ a universal enveloping morphism. By universality, two universal enveloping groups ( G 1 , τ 1 ) and ( G 2 , τ 2 ) 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 ) := i j I G i j | all relations in G i j , and ϕ i j j ( x ) = ϕ k j j ( x ) for all i j k I and all x U

and where τ ^ = { τ ^ i j i j I } with the canonical homomorphism τ ^ i j : G i j G ( 𝒜 ) for all i j I . The canonical universal enveloping group of a U-amalgam is a universal enveloping group (cf. [25, Lemma 1.3.2]).

Lemma 7

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

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

  • 𝒜 = { G i j , ϕ i j i i j I } is a V-amalgam over I,

  • α = { π , ρ i , α i j i j I } is an amalgam epimorphism 𝒜 ~ 𝒜 ,

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

Then the following hold:

  1. (a)

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

  2. (b)

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

  3. (c)

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

Proof.

(a) Let i j I . Since α i j is an epimorphism, we must have

τ ~ i j := τ π ( i ) π ( j ) α i j

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

τ ~ i j ( G ~ i j ) = ( τ π ( i ) π ( j ) α i j ) ( G ~ i j ) = τ π ( i ) π ( j ) ( G π ( i ) π ( j ) ) ,

and so

G = τ i j ( G ~ i j ) = τ ~ i j ( G ~ i j ) .

Moreover, for i j k I we find

τ ~ i j ϕ ~ i j j = τ π ( i ) π ( j ) α i j ϕ ~ i j j
= τ π ( i ) π ( j ) ϕ π ( i ) π ( j ) π ( j ) ρ π ( j )
= τ π ( k ) π ( j ) ϕ π ( k ) π ( j ) π ( j ) ρ π ( j )
= τ π ( k ) π ( j ) α k j ϕ ~ k j j
= τ ~ k j ϕ ~ k j j .

Hence ( G , { τ ~ i j } ) 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 G π ( i ) π ( j ) , G and G ~ i j , G ~ . ∎

Notation 8

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 } .

Proposition 9

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

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

    G ~ i j = ϕ ~ i j i ( U ~ ) , ϕ ~ i j j ( U ~ ) ,

  • 𝒜 = { G i j , ϕ i j i i j I } is a U-amalgam over I,

  • α = { π , ρ i , α i j i j I } is an amalgam epimorphism 𝒜 ~ 𝒜 ,

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

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

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

    as in Lemma 7(b).

For i , j I , with i j define Z i j i := ϕ ~ i j i ( V ~ ) and Z i j ϕ ~ := Z i j i , Z i j j , as well as A i j := ker ( α i j ) . Then if

A i j Z i j ϕ ~ Z ( G ~ i j ) ,

it follows that G ~ is a central extension of G by N := τ ~ i j ( A i j ) i j I .

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:

  1. (a)

    N Z ( G ~ ) ,

  2. (b)

    G ~ / N G .

Consider the following commutative diagram:

For i j I set

Z i := τ ~ i j ( Z i j i ) = ( τ ~ i j ϕ ~ i j i ) ( V ~ ) and G ~ i := ( τ ~ i j ϕ ~ i j i ) ( U ~ ) .

The hypothesis implies

τ ~ i j ( G ~ i j ) = G ~ i , G ~ j and G ~ = τ ~ i j ( G ~ i j ) i j I = G ~ i i I .

Moreover,

Z i = τ ~ i j ( Z i j i ) τ ~ i j ( Z ( G ~ i j ) ) Z ( τ ~ i j ( G ~ i j ) ) = Z ( G ~ i , G ~ j ) ,

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

Z i i I Z ( G ~ i i I ) = Z ( G ~ ) .

Therefore

N = τ ~ i j ( A i j ) i j I τ ~ i j ( Z i j i ) i j I = Z i i I Z ( G ~ ) ,

i.e., (a) holds.

Commutativity of the diagram implies N ker ( α ^ ) and so the homomorphism theorem yields an epimorphism G ~ / N G , g N α ^ ( 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 g G i j , let g ~ α i j - 1 ( g ) . Then

τ ~ i j ( α i j - 1 ( g ) ) = τ ~ i j ( g ~ A i j ) τ ~ i j ( g ~ ) N = τ ~ i j ( α i j - 1 ( g ) ) N G ~ / N .

Thus one obtains a well-defined homomorphism

τ ^ π ( i ) π ( j ) : G π ( i ) π ( j ) G ~ / N , g τ ~ i j ( α i j - 1 ( g ) ) N .

Then ( G ~ / N , { τ ^ i j } ) is an enveloping group for 𝒜 . In particular, for u U and i j k I one has

( τ ^ π ( i ) π ( j ) ϕ π ( i ) π ( j ) π ( j ) ) ( u ) = τ ~ i j ( α i j - 1 ( ϕ π ( i ) π ( j ) π ( j ) ( u ) ) )
= τ ~ i j ( ϕ ~ i j j ( ( ρ j ) - 1 ( u ) ) )
= τ ~ k j ( ϕ ~ k j j ( ( ρ j ) - 1 ( u ) ) )
= τ ~ k j ( α k j - 1 ( ϕ π ( k ) π ( j ) π ( j ) ( u ) ) )
= ( τ ^ π ( k ) π ( j ) ϕ π ( k ) π ( j ) π ( j ) ) ( u ) .

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

β τ i j = τ ^ i j .

By the definition of α ^ and τ ^ i j one finds

α ^ τ ^ i j = τ i j .

Therefore

( β α ^ ) τ ^ i j = τ ^ i j and ( α ^ β ) τ i j = τ i j .

But ( G , τ ) and ( G ~ , τ ~ ) are universal enveloping groups; their uniqueness property implies that β α ^ = id G ~ / N and α ^ β = id G and hence as claimed G ~ / N G . We have shown assertion (b). ∎

4 Cartan matrices and Dynkin diagrams

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

Definition 1

Let I be a non-empty set. A generalised Cartan matrix over I is a matrix A = ( a ( i , j ) ) i , j I such that for all i j I ,

  1. (a)

    a ( i , i ) = 2 ,

  2. (b)

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

  3. (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 , j I with i j .

Definition 2

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 v w . Let E 0 ( Π ) := ( V ( Π ) 2 ) E ( Π ) , and let E 1 ( Π ) , E 2 ( Π ) , E 3 ( Π ) , resp. E ( Π ) be the subsets of E ( Π ) of edges of valency 1, 2, 3, resp. . The elements of E 1 ( Π ) , E 2 ( Π ) , E 3 ( Π ) are called edges of type A 2 , C 2 resp. G 2 . The diagram Π is irreducible if it is connected as a graph, it is simply laced if all edges have valency 1, it is doubly laced if all edges have valency 1 or 2, and it is two-spherical if no edge has valency . If V ( Π ) is finite, then a labelling of Π is a bijection σ : I V ( Π ) , where I := { 1 , , | V ( Π ) | } .

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

Remark 3

Let I be a non-empty set and A = ( a ( i , j ) ) i , j I 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 , j I with i j , there is an edge between i and j if and only if a ( i , j ) 0 . The valency of the edge then is v i j := a ( i , j ) a ( j , i ) { 1 , 2 , 3 } . If v i j > 1 , then the edge is directed i j if and only if a ( i , j ) = - v i j < - 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 , j I over I := V by setting for i j I ,

a ( i , i ) := 2 , a ( i , j ) := { 0 , if { i , j } E ( Π ) , - 2 , if { i , j } E 2 ( Π ) and i j , - 3 , if { i , j } E 3 ( Π ) and i j , - 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.

Notation 4

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 i j , if a ( i , j ) is even, and i j , 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.

Definition 1

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

O ( q ) := { a GL ( V ) q ( a v ) = q ( v ) for all v V } ,
SO ( q ) := O ( q ) SL ( V ) .

Given n , let q n : n , x i = 1 n x i 2 be the standard quadratic form on n , and

O ( n ) := { a GL n ( ) a a t = E n } O ( q n ) = O ( - q n ) ,
SO ( n ) := O ( n ) SL n ( ) SO ( q n ) = SO ( - q n ) O ( q n ) = O ( - q n ) .

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

Notation 2

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

I := { e i i I } , V I := I n , q I := q n | V I : V I .

There are canonical isomorphisms

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

and

M I : End ( V I ) M | I | ( ) , a M I ( 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

ε I : O ( q I ) O ( q n ) ,

inducing a canonical embedding

M ε I M I - 1 : O ( | I | ) O ( n ) ,

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

ε i j : 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].

Definition 1

Let ( , V , q ) be a quadratic space and let T ( V ) = n 0 V n be the tensor algebra of V. The identity V 0 = provides a ring monomorphism T ( V ) , and the identity V 1 = V a vector space monomorphism V T ( V ) ; these allow one to identify , V with their respective images in T ( V ) . For

( q ) := v v - q ( v ) v V

define the Clifford algebra of q as

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

Moreover, let

Cl ( q ) * := { x Cl ( q ) there exists y Cl ( q ) such that x y = 1 } .

The transposition map is the involution

τ : Cl ( q ) Cl ( q ) induced by v 1 v k v k v 1 , v i V ,

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

Π : Cl ( q ) Cl ( q ) given by v 1 v k ( - 1 ) k v 1 v k , v i V ,

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

Cl 0 ( q ) := { x Cl ( q ) Π ( x ) = x }

and

Cl 1 ( q ) := { x Cl ( q ) Π ( x ) = - x } ,

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

Cl ( q ) = Cl 0 ( q ) Cl 1 ( q )

and

Cl i ( q ) Cl j ( q ) Cl i + j ( q ) for i , j 2 .

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

σ : Cl ( q ) Cl ( q ) , x x ¯ := τ Π ( x ) = Π τ ( x ) ,

and the spinor norm

N : Cl ( q ) Cl ( q ) , x x x ¯ .

Notation 2

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

Definition 3

Given x Cl ( q ) * , the map

ρ x : Cl ( q ) Cl ( q ) , y Π ( x ) y x - 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 ) := { x Cl ( q ) * ρ x ( v ) V for all v V }

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 ) , x ρ x ,

which is the twisted adjoint representation.

Definition 4

Given n and V = n , we set

Cl ( n ) := Cl ( - q n ) and Γ ( n ) := Γ ( - q n ) .

Recall that q n 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.

Remark 5

(a) Let n and let e 1 , , e n be the standard basis of n . Then the following hold in Cl ( n ) for 1 i j n ,

e i 2 = - 1 ,
e i e j = - e j e i ,
( e i e j ) 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

( q n ) ( e i + e j ) ( e i + e j ) - q ( e i + e j )
= e i e i + e i e j + e j e i + e j e j - q ( e i ) - q ( e j ) - 2 b ( e i , e j )
= e i e j + e j e i ,

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

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

Lemma 6

The map N : Cl ( q ) Cl ( q ) induces a homomorphism

N : Γ ( q ) *

such that

N ( Π ( x ) ) = N ( x ) for all x Γ ( q ) .

Proof.

Cf. [13, Proposition 1.9]. ∎

Definition 7

The group

Pin ( q ) := { x Γ ( q ) N ( x ) = 1 } Γ ( q )

is the pin group with respect to q , and

Spin ( q ) := Pin ( q ) Cl 0 ( 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 ( - q n ) and Spin ( n ) := Spin ( - q n ) .

Theorem 8

The following hold:

  1. (a)

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

  2. (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

    ρ n := M ρ : Pin ( n ) O ( n ) , ρ n := M ρ : Spin ( n ) SO ( n )

    with ker ( ρ n ) = { ± 1 } in both cases.

  3. (c)

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

Proof.

See [13, Theorem 1.11]. ∎

Remark 9

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

(b) Let H 1 Spin ( n ) and H 2 Pin ( n ) be such that - 1 H 1 and - 1 H 2 , respectively, and let H ~ i := ρ n ( H i ) . Then we have H i = ρ n - 1 ( H ~ i ) . We will explicitly determine these groups for some canonical subgroups of SO ( n ) and O ( n ) .

Lemma 10

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

{ e i i I } { e i 1 i n }

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

Pin ( - q I ) Pin ( - q n )

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 ~ i j : Spin ( 2 ) Spin ( n ) .

Proof.

Let x Γ ( - q I ) . By definition,

ρ ε ~ I ( x ) ( v ) = ( ε I ρ ) ( v ) V I V = n for all v V I .

Since e i e j = - e j e i for all i j I by Remark 5 (a), for each -basis vector y = e j 1 e j k of ε ~ I ( Cl ( - q I ) ) and all i { 1 , , n } \ I one has

Π ( y ) e i = e i y .

Hence

Π ( ε ~ I ( x ) ) e i = e i ε ~ I ( x )

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

ρ ε ~ I ( x ) ( e i ) = Π ( ε ~ I ( x ) ) e i ε ~ I ( x ) - 1 = e i n .

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

ρ ε ~ I = ε I ρ .

Therefore ε I ( x ) Γ ( - q n ) . Finally,

N ( ε ~ I ( x ) ) = ε ~ I ( N ( x ) ) = ε ~ I ( 1 ) = 1 ,

whence

ε ~ I ( x ) Pin ( - q n ) .

Remark 11

Since ε ~ I ( Spin ( - q I ) ) = e i e j i j I Cl 0 ( - q n ) , one has

ε ~ I ( Spin ( - q I ) ) Pin ( - q n ) Cl 0 ( - q n ) = Spin ( - q n ) .

Consequence 12

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

ρ n - 1 ( ε I ( O ( | I | ) ) ) = ε ~ I ( Pin ( - q I ) ) 𝑎𝑛𝑑 ρ n - 1 ( ε I ( SO ( | I | ) ) ) = ε ~ I ( Spin ( - q I ) ) .

Proof.

By Lemma 10, one has

ρ n ε ~ I ( Pin ( - q I ) ) = ε I ( O ( | I | ) ) and ρ n ε ~ I ( Spin ( - q I ) ) = ε I ( SO ( | I | ) ) ,

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

Remark 13

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

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

id ε ~ I i : Pin ( m ) Pin ( n )

and ε I for the map

M ε ε I M ε I - 1 : O ( m ) O ( n ) .

Consequently, we obtain the following commutative diagram:

Remark 14

According to [13, Corollary 1.12], the group Pin ( n ) is generated by the set { v n N ( 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 g Spin ( 2 ) is of the form

g = i = 1 2 k ( a i e 1 + b i e 2 )
= i = 1 k ( ( a 2 i - 1 a 2 i + b 2 i - 1 b 2 i ) + ( a 2 i - 1 b 2 i - a 2 i b 2 i - 1 ) e 1 e 2 )
= : a + b e 1 e 2 .

The requirement a i e 1 + b i e 2 { v n N ( v ) = 1 } is equivalent to

a i 2 + b i 2 = ( a i e 1 + b i e 2 ) ( - a i e 1 - b i e 2 )
= ( a i e 1 + b i e 2 ) ( a i e 1 + b i e 2 ) ¯
= N ( a i e 1 + b i e 2 ) = 1 .

Moreover,

1 = N ( g )
= N ( a + b e 1 e 2 )
= ( a + b e 1 e 2 ) ( a + b e 1 e 2 ) ¯
= ( a + b e 1 e 2 ) ( a + b e 2 e 1 ) = a 2 + b 2 .

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

Spin ( 2 ) = { cos ( α ) + sin ( α ) e 1 e 2 α } .

One has

( cos ( α ) + sin ( α ) e 1 e 2 ) - 1 = cos ( α ) - sin ( α ) e 1 e 2 = cos ( - α ) + sin ( - α ) e 1 e 2 ,

i.e., the map

Spin ( 2 ) , α cos ( α ) + sin ( α ) e 1 e 2

is a group homomorphism from the real numbers onto the circle group. The twisted adjoint representation ρ 2 maps the element cos ( α ) + sin ( α ) e 1 e 2 Spin ( 2 ) to the transformation

x 1 e 1 + x 2 e 2 ( cos ( α ) + sin ( α ) e 1 e 2 ) ( x 1 e 1 + x 2 e 2 ) ( cos ( α ) - sin ( α ) e 1 e 2 )
= x 1 ( cos ( α ) 2 - sin ( α ) 2 ) e 1 - 2 x 2 cos ( α ) sin ( α ) e 1
+ 2 x 1 cos ( α ) sin ( α ) e 2 + x 2 ( cos ( α ) 2 - sin ( α ) 2 ) e 2
= ( x 1 cos ( 2 α ) - x 2 sin ( 2 α ) ) e 1 + ( x 1 sin ( 2 α ) + x 2 cos ( 2 α ) ) e 2 ,

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

( cos ( 2 α ) - sin ( 2 α ) sin ( 2 α ) cos ( 2 α ) ) SO ( 2 ) .

In other words, ρ 2 is the double cover of the circle group by itself, cf. Theorem 8 (b).

Similarly, each element g Spin ( 3 ) is of the form

g = i = 1 2 k ( a i e 1 + b i e 2 + c i e 3 ) = a + b e 1 e 2 + c e 2 e 3 + d e 3 e 1

and each element h Spin ( 4 ) of the form

h = i = 1 2 k ( a i e 1 + b i e 2 + c i e 3 + d i e 4 )
= h 1 + h 2 e 1 e 2 + h 3 e 2 e 3 + h 4 e 3 e 1 + h 5 e 1 e 2 e 3 e 4
+ h 6 e 4 e 3 + h 7 e 4 e 1 + h 8 e 4 e 2 .

7 The isomorphism 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.

Definition 1

Denote by

:= { a + b i + c j + d k a , b , c , d }

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

¯ : , x = a + b i + c j + d k x ¯ = a - b i - c j - d k

be the standard involution, and let

U 1 ( ) := { x x x ¯ = 1 }

be the group of unit quaternions.

Remark 2

By [13, Section 1.4] one has

Spin ( 3 ) U 1 ( ) and Spin ( 4 ) Spin ( 3 ) × Spin ( 3 ) U 1 ( ) × U 1 ( ) .

The isomorphism Spin ( 3 ) U 1 ( ) in fact is an immediate consequence of the isomorphism Cl 0 ( 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 ) U 1 ( ) × U 1 ( ) can be described as follows (see [13, Section 1.4]). By Remark 14 each element of Spin ( 4 ) is of the form

a + b e 1 e 2 + c e 2 e 3 + d e 3 e 1 + a e 1 e 2 e 3 e 4 + b e 4 e 3 + c e 4 e 1 + d e 4 e 2 .

For

i := e 1 e 2 , j := e 2 e 3 , k := e 3 e 1 , 𝕀 := e 1 e 2 e 3 e 4 ,
i := e 4 e 3 , j := e 4 e 1 , k := e 4 e 2

one has

i j = k , j k = i , k i = j ,
i 𝕀 = 𝕀 i = i , j 𝕀 = 𝕀 j = j , k 𝕀 = 𝕀 k = k ,
i 2 = j 2 = k 2 = - 1 , 𝕀 2 = 1 , σ ( 𝕀 ) = 𝕀 ,

where σ ( 𝕀 ) denotes the Clifford conjugate of 𝕀 , cf. Definition 1. We conclude that for every x Spin ( 4 ) there exist uniquely determined u = a + b i + c j + d k , v = a + b i + c j + d k such that

x = u + 𝕀 v .

One computes

N ( x ) = N ( u + 𝕀 v ) = ( u + 𝕀 v ) ( u ¯ + 𝕀 v ¯ ) = u u ¯ + v v ¯ + 𝕀 ( u v ¯ + v u ¯ ) ,

i.e.,

N ( x ) = 1 u u ¯ + v v ¯ = 1 and u v ¯ + v u ¯ = 0 .

Hence, for 1 = N ( x ) = N ( u + 𝕀 v ) , 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 + 𝕀 v ( u + v , u - v )

is a well-defined bijection and, since

( u + 𝕀 v ) ( u + 𝕀 v ) = u u + v v + 𝕀 ( u v + v u )

and

( u + v , u - v ) ( u + v , u - v ) = ( u u + v v + u v + v u , u u + v v - ( u v + v u ) ) ,

it is in fact an isomorphism of groups.

Consequently, there exists a group epimorphism

η ~ : Spin ( 4 ) Spin ( 3 ) , u + 𝕀 v u + v .

Remark 3

Using this isomorphism

Spin ( 4 ) Spin ( 3 ) × Spin ( 3 ) U 1 ( ) × U 1 ( )

there exists a natural homomorphism

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

Note that the restrictions ( a , 1 ) ( x a x ) and ( 1 , b ) ( x x b - 1 ) are both injections of Spin ( 3 ) U 1 ( ) 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

, x a x b - 1 for a , b U 1 ( ) ;

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 ( x a x a - 1 )

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

, x a x a - 1 for a U 1 ( ) ;

see also [37, Lemma 11.24].

Remark 4

There also exists a group epimorphism

η : SO ( 4 ) SO ( 3 )

induced by the map

SO ( 4 ) { , x a x b - 1 a , b U 1 ( )
{ , x a x a - 1 a U 1 ( ) } SO ( 3 )
( x a x b - 1 ) ( x a x a - 1 ) .

Altogether, one obtains the following commutative diagram:

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

Notation 1

For SO ( 2 ) × SO ( 2 ) = { ( a , b ) a , b SO ( 2 ) } let

ι 1 : SO ( 2 ) SO ( 2 ) × SO ( 2 ) , x ( x , 1 ) ,

and

ι 2 : SO ( 2 ) SO ( 2 ) × SO ( 2 ) , x ( 1 , x ) .

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

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

Hence

( ρ 2 × ρ 2 ) ι ~ 1 = ι 1 ρ 2 , ( ρ 2 × ρ 2 ) ι ~ 2 = ι 2 ρ 2 .

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 ρ 2 × ρ 2 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 ) e 1 e 2 Spin ( 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 ρ 2 from Theorem 8 satisfies ρ 2 = sq ψ - 1 , i.e.

ρ 2 : Spin ( 2 ) SO ( 2 ) , S ( α ) D ( 2 α ) .

Proposition 2

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

Proof.

Define γ ~ := ψ γ ψ - 1 . Then

ρ 2 γ ~ = ( sq ψ - 1 ) ( ψ γ ψ - 1 )
= sq γ ψ - 1
= γ sq ψ - 1
= γ ρ 2 .

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

Corollary 3

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

( ρ 2 × ρ 2 ) γ ~ = γ ( ρ 2 × ρ 2 ) .

Proof.

Let γ ~ := ψ γ ψ - 1 , where

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

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

Proposition 4

Let n 3 . Given an automorphism γ Aut ( SO ( n ) ) , there is a unique automorphism γ ~ Aut ( Spin ( n ) ) such that

ρ n γ ~ = γ ρ n .

Proof.

For n 3 , 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

γ ρ n = ρ n γ ~ and γ - 1 ρ n = ρ n γ ~ .

Hence

ρ n γ ~ γ ~ = γ ρ n γ ~ = γ γ - 1 ρ n = ρ n

and, similarly,

ρ n γ ~ γ ~ = ρ n .

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 ε i j in Notation 2 and of ε ~ i j in Lemma 10.

Proposition 5

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

ker ( ρ n ϕ ) = { 1 , - 1 } 𝑎𝑛𝑑 ρ n ϕ = ε i j ρ 2

for some i j I . Then ϕ = ε ~ i j .

Proof.

By Consequence 12 one has

ϕ ( Spin ( 2 ) ) ( ρ n - 1 ε i j ρ 2 ) ( Spin ( 2 ) ) = ρ n - 1 ( ε i j ( SO ( 2 ) ) ) = ε ~ i j ( Spin ( 2 ) ) .

By hypothesis, ker ϕ { 1 , - 1 } . If - 1 ker ϕ , then

1 = ϕ ( - 1 ) = ϕ ( S ( π ) ) = ϕ ( S ( π 2 ) ) 2 ,

i.e., we have ϕ ( S ( π 2 ) ) { 1 , - 1 } , whence S ( π 2 ) ker ( ρ n ϕ ) , a contradiction. Consequently, ϕ is a monomorphism.

Consider the following commuting diagram:

One has ρ 2 ε ~ i j - 1 ϕ = ρ 2 = id ρ 2 . Since ϕ is injective, the map ε ~ i j - 1 ϕ is an automorphism of Spin ( 2 ) . Hence Proposition 2 implies ε ~ i j - 1 ϕ = 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 ε 12 , ε 23 : SO ( 2 ) SO ( 3 ) from Notation 2 and the maps ι 1 , ι 2 : SO ( 2 ) SO ( 2 ) × SO ( 2 ) from Notation 1.

Definition 1

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

𝒜 = { G i j , ϕ i j i i j I }

such that

G i j = { SO ( 3 ) , if { i , j } σ E ( Π ) , SO ( 2 ) × SO ( 2 ) , if { i , j } σ E ( Π ) , for all i j I

and for i < j I ,

ϕ i j i ( SO ( 2 ) ) = { ε 12 ( SO ( 2 ) ) , if { i , j } σ E ( Π ) , ι 1 ( SO ( 2 ) ) , if { i , j } σ E ( Π ) ,
ϕ i j j ( SO ( 2 ) ) = { ε 23 ( SO ( 2 ) ) , if { i , j } σ E ( Π ) , ι 2 ( SO ( 2 ) ) , if { i , j } σ E ( Π ) .

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

𝒜 ( Π , σ , SO ( 2 ) ) := { G i j , ϕ i j i i j I }

with respect to Π and σ with

ϕ i j i = { ε 12 , if { i , j } σ E ( Π ) , ι 1 , if { i , j } σ E ( Π ) ,
ϕ i j j = { ε 23 , if { i , j } σ E ( Π ) , ι 2 , if { i , j } σ E ( Π ) ,

for all i < j I .

Remark 2

The key difference between the standard SO ( 2 ) -amalgam and an arbitrary SO ( 2 ) -amalgam 𝒜 = { G i j , ϕ i j i i j I } with respect to Π and σ is that, for instance, ε 12 - 1 ϕ i j i 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 ε 12 - 1 ϕ i j i 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.

Convention 3

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.

Lemma 4

Let

D := ( 1 - 1 1 ) SO ( 3 ) .

Then the map γ D : SO ( 3 ) SO ( 3 ) , A D A D - 1 = D A D is an automorphism of SO ( 3 ) such that

γ D ε 12 = ε 23 𝑎𝑛𝑑 γ D ε 23 = ε 12 .

Proof.

Given ( x y - y x ) SO ( 2 ) , we have

( 1 - 1 1 ) ( x y - y x 1 ) ( 1 - 1 1 ) = ( 1 y - x x y ) ( 1 - 1 1 )
= ( 1 x y - y x ) .

The second assertion follows analogously. ∎

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

Consequence 5

Let Π be a simply-laced diagram with labellings σ 1 , σ 2 : I V . Then

𝒜 (