weyl_short.pdf

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WEYL GROUPS OF SOME HYPERBOLIC KAC-MOODY ALGEBRAS

ALEX J. FEINGOLD AND DANIEL VALLIERES `

Abstract. We use the theory of Clifford algebras and Vahlen groups to study Weyl groups of

hyperbolic Kac-Moody algebras T

n , obtained by a process of double extension from a Cartan

matrix of finite type Tn, whose corresponding generalized Cartan matrices are symmetric.

Contents

1. Introduction 1

2. Generalities on orthogonal geometries 2

3. Clifford algebras, Pin and Spin groups 5

3.1. The Clifford group 7

3.2. Abstract Pin and Spin groups 8

3.3. Pin and Spin groups 9

3.4. Lorentzian geometry over R 9

3.5. Change of fields 10

4. Vahlen groups 11

4.1. Vahlen groups 11

4.2. Change of fields 15

5. Generalized Cartan matrices, system of simple roots and Weyl groups 15

5.1. A useful normalization 19

5.2. Canonical Lorentzian extensions 19

5.3. Change of fields 21

6. Weyl groups of the hyperbolic canonical Lorentzian extensions T

n with symmetric Cartan

matrices 21

6.1. Spinor norm of outer autormorphisms 24

7. Connections with previous descriptions of the Weyl group 24

7.1. Paravectors 24

7.2. Vahlen groups for paravectors 26

7.3. Hermitian matrices 30

References 31

1. Introduction

In [12], Feingold and Frenkel gained significant new insight into the structure of a particularly

interesting rank 3 hyperbolic Kac–Moody algebra which they called F (also known as A

), along

with some connections to the theory of Siegel modular forms of genus 2. The first vital step in

their work was the discovery that the even part of the Weyl group of that Kac-Moody algebra is

SW(F) ∼= P SL(2, Z). (If W is a Weyl group, we will denote its even part by SW.)

In [13], a coherent picture of Weyl groups was presented for many higher rank hyperbolic Kac–

Moody algebras using lattices and subrings of the four normed division algebras. Specifically, the

Weyl groups of all hyperbolic algebras of ranks 4, 6 and 10 which can be obtained by a process of

Date: January 16, 2016.

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2 ALEX J. FEINGOLD AND DANIEL VALLIERES `

double extension, admit realizations in terms of generalized modular groups over the complex numbers

C, the quaternions H, and the octonions O, respectively. In particular, the authors found in the rank

four situation that the even part of the Weyl groups of the Kac-Moody algebra A

is the Bianchi

group P SL(2, O−3), where O−3 the ring of integers of Q(

√

−3).

One could ask if there is a similar phenomenon for all the hyperbolic Kac-Moody algebras T

where Tn is any of the finite type root system, but it is not clear what to take instead of a normed

division algebra. In [15], the authors used the quaternions and the octonions in their study of some

Weyl groups. In this paper, we adopt another approach. We use the theory of Vahlen groups and

Clifford algebras in order to study the Weyl groups of the hyperbolic Kac-Moody algebras T

n whose

Cartan matrices are symmetric.

Our paper is organized as follows. In §2, we remind the reader about generalities on orthogonal

geometries. Then, we gather some results on Clifford algebras, Pin and Spin groups which we shall

need later in §3. Section §4 introduces Vahlen groups. In the literature, Vahlen groups have been

defined for the paravector case as well as for the non-paravector case. In §4, we place ourselves in the

non-paravector case, whereas the paravector situation is treated in §7.1 and §7.2. Section §5 contains

a useful, though very simple, introduction to generalized Cartan matrices, systems of simple roots

and Weyl groups. The core of this paper in contained in §6, where we give a description of several

Weyl groups. At last, we explain in §7 the connections between our approach and the one adopted

previously in [13].

Acknowledgements: We would like to thank Igor Frenkel for suggesting this direction of research.

AJF gratefully acknowledges the hospitality of the Albert Einstein Institute on various visits, and the

IHES. We would also like to thank Joel Dodge for various very useful discussions at the beginning of

this project.

2. Generalities on orthogonal geometries

Throughout this paper, F will denote a field with characteristic different from 2. In fact, all the

fields considered in this paper have characteristic zero. The results of this section are well-known and

we will not repeat the proofs. We refer the reader to [7] and [10].

Let V be a finite dimensional F-vector space of dimension n. If V is equipped with a symmetric

F-bilinear form S : V × V −→ F, then we say that (V, S) is an orthogonal geometry. If the symmetric

F-bilinear form is clear from the context, we might drop it from the notation, and we will refer to an

orthogonal geometry just by V .

Instead of working with the symmetric F-bilinear form S, one can work with the associated quadratic

form given by q(v) = S(v, v) for all v ∈ V . A pair (V, q), where V is a finite dimensional vector space

over F and q is a quadratic form on V is called a quadratic space. We have a one-to-one correspondence

between symmetric F-bilinear forms S and quadratic forms q. Given a quadratic form q, one recovers

S via the formula

S(v1, v2) = 1

(q(v1 + v2) − q(v1) − q(v2)),

whenever v1, v2 ∈ V .

Given an orthogonal geometry V , the radical of V , denoted by Rad(V ), is defined as usual, i.e. it

is the kernel of the linear transformation V −→ V

∗ defined by v 7→ S(v, ·). We remind the reader of

the following important definition.

Definition 2.1. Let V be an orthogonal geometry. Then,

(1) V is called non-singular if Rad(V ) = 0,

(2) V is called isotropic if Rad(V ) = V ,

(3) A vector v ∈ V is called isotropic if q(v) = 0. Otherwise, it is called non-isotropic.

It is simple to check that an orthogonal geometry V is isotropic if and only if every vector v ∈ V is

isotropic.

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WEYL GROUPS OF SOME HYPERBOLIC KAC-MOODY ALGEBRAS 3

Let (V1, S1) and (V2, S2) be two orthogonal geometries. A linear transformation f : V1 −→ V2 is

called an orthogonal map if

S2(f(v1), f(v

)) = S1(v1, v0

for all v1, v0

1 ∈ V1. An orthogonal map f : V1 −→ V2 is called an isometry if there exists an orthogonal

map g : V2 −→ V1 satisfying f ◦ g = idV2

and g ◦ f = idV1

. Note that if f : V1 −→ V2 is a bijective

orthogonal map, then it is an isometry. More generally, an F-linear transformation f : V1 −→ V2 is

called an orthogonal similitude if there exists λ ∈ F

× such that

S2(f(v1), f(v

)) = λS1(v1, v0

for all v1, v0

1 ∈ V1. The constant λ ∈ F

× is called the factor of similitude of f. It is simple to check that

if f : V1 −→ V2 is an orthogonal similitude between two orthogonal geometries and V1 is non-singular,

then f is necessarily injective.

The set of isometries of an orthogonal geometry V into itself forms a subgroup of the general linear

group GL(V ) which is denoted by O(V, S) or O(V ) if S is understood from the context. Moreover, we

let GO(V ) be the group of orthogonal similitudes, that is

GO(V ) = {g ∈ GL(V )| S(g(v1), g(v2)) = λ(g)S(v1, v2) for some λ(g) ∈ F

×}.

Note that the map λ : GO(V ) −→ F

× is a group morphism and O(V ) = ker(λ). If V is an orthogonal

geometry over F with symmetric F-bilinear form S and λ ∈ F

×, then we let V

λ be the orthogonal

geometry obtained from V by rescaling the symmetric F-bilinear form S by a factor λ. That is, the

vector space V is the same, but the pairing is given by λ · S instead of S.

If V is non-singular, then it is simple to check that any σ ∈ O(V ) satisfies det(σ) = ±1. The ones

satisfying det(σ) = 1 are called rotations and they form a subgroup of O(V ) which is denoted by

SO(V, S) or more simply by SO(V ).

If U is a subspace of V , then the symmetric F-bilinear form on V induces by restriction an orthogonal

geometry on U, and we have

(1) Rad(U) = U ∩ U

⊥.

If V is an orthogonal sum of finitely many subspaces, i.e. V = U1 ⊥ . . . ⊥ Ur, then

(2) Rad(V ) = Rad(U1) ⊥ . . . ⊥ Rad(Ur),

and thus V is non-singular if and only if Ui

is non-singular for all i = 1, . . . , r.

Assuming now that V is a non-singular orthogonal geometry and U is a subspace of V , we have

(1) U = (U

⊥)

⊥,

(2) Rad(U

⊥) = U ∩ U

⊥,

(3) dim(U) + dim(U

⊥) = dim(V ).

Hence, combined with (1) and (2), we obtained the following result.

Proposition 2.2. Let V be a non-singular orthogonal geometry. The subspace U is non-singular if

and only if there exists another subspace W of V satisfying V = U ⊥ W.

Note that if L = Span(v) is a line (v 6= 0), then L is non-singular if ond only if v is a non-isotropic

vector.

Definition 2.3. An orthogonal geometry is called irreducible if it cannot be written as an orthogonal

sum of proper subspaces.

If V is an orthogonal geometry and V = Rad(V ) ⊕ U, then we have V = Rad(V ) ⊥ U. Hence, if V

is irreducible, then V is either non-singular or isotropic. Moreover, an irreducible subspace necessarily

has dimension one. In other words, an irreducible subspace of an orthogonal geometry is a line. A

standard argument gives us the following well-known result.

Proposition 2.4. If V is an orthogonal geometry, then V = L1 ⊥ . . . ⊥ Lr, where Li = Span(vi) are

lines. Moreover, V is non-singular if and only if vi is a non-isotropic vector for all i = 1, . . . , r.

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4 ALEX J. FEINGOLD AND DANIEL VALLIERES `

The set {v1, . . . , vr} of the last proposition is called an orthogonal basis. We will now recall the

definition of some important isometries in O(V ). An isometry σ ∈ O(V ) is called an involution if

2 = 1. If σ is an involution, then we let

U =

1 − σ

V and W =

1 + σ

(Recall that we are staying away from characteristic 2.) It is then a simple matter to show that

V = U ⊥ W and σ = −idU ⊥ idW . The dimension of U is called the type of σ. Note that by

Proposition 2.2, U and W are non-singular subspaces. An involution of type 1 is called a symmetry

with respect to the hyperplane W or less precisely an hyperplane reflection or even more simply a

reflection.

If σ = −idL ⊥ idH is an hyperplane reflection and v ∈ L is a non-zero vector, then v is a non- isotropic vector, since L is non-singular. On the other hand, if we start with a non-isotropic vector

v ∈ V , then it is simple to check that rv ∈ O(V ) given by

rv(w) = w − 2 ·

S(w, v)

S(v, v)

whenever w ∈ V , is a symmetry with respect to the hyperplane L

⊥ where L = Span(v). Conversely,

every hyperplane reflection −idL ⊥ idH is of the form rv for some non-isotropic vector v ∈ L. Theorem

2.6 below is fundamental, but we first need the following lemma whose proof is left to the reader.

Lemma 2.5. Let V be an orthogonal geometry and let v, w ∈ V . If q(v) = q(w) 6= 0, then either

(1) q(v − w) 6= 0,

(2) q(v + w) 6= 0.

In case (1), we have rv−w(v) = w, and in case (2), we have rw ◦ rv+w(v) = w.

We can now show:

Theorem 2.6. Let V be a non-singular orthogonal geometry. Then, every σ ∈ O(V ) is a product of

hyperplane reflections.

Proof. Let σ ∈ O(V ). By Proposition 2.4, we know that V has an orthogonal basis, that is

V = F e1 ⊥ . . . ⊥ F en,

for some non-isotropic vectors ei ∈ V . Define ψi ∈ O(V ) inductively as follows:

ψi =

(

rψi−1·...·ψ1·σ(ei)−ei

, if q(ψi−1 · . . . ψ1 · σ(ei) − ei) 6= 0;

rei

◦ rψi−1·...·ψ1·σ(ei)+ei

, otherwise.

One checks using Lemma 2.5 that ψiψi−1 . . . ψ1σ(ej ) = ej for all j = 1, . . . , i. It follows that σ is a

product of hyperplane reflections, and this is what we wanted to show.

We remark that there is a strengthening of Theorem 2.6 called the Cartan-Dieudonn ́e theorem which

stipulates that if the dimension of V is n, then every σ ∈ O(V ) is a product of at most n hyperplane

reflections, but we shall not make use of this fact in this paper.

Assume now that F is an ordered field. (This implies in particular that the characteristic of F is

zero.) We remind the reader of the following definition.

Definition 2.7. Let V be an orthogonal geometry. Then, the geometry is called

(1) positive semi-definite if S(v, v) ≥ 0 for all v ∈ V ,

(2) negative semi-definite if S(v, v) ≤ 0 for all v ∈ V ,

(3) positive definite if S(v, v) > 0 for all v ∈ V r {0},

(4) negative definite if S(v, v) < 0 for all v ∈ V r {0}.