Hyperbolic Weyl Groups and the Four Normed Division Algebras

Alex Feingold1, Axel Kleinschmidt2, Hermann Nicolai3

1 Department of Mathematical Sciences, State University of New York Binghamton, New York 13902–6000, U.S.A.

2 Physique Theorique´ et Mathematique,´ Universite´ Libre de Bruxelles & International Solvay Institutes, Boulevard du Triomphe, ULB – CP 231, B-1050 Bruxelles, Belgium

3 Max-Planck-Institut fur¨ Gravitationsphysik, Albert-Einstein-Institut Am Muhlenberg¨ 1, D-14476 Potsdam, Germany

– p. Introduction

Presented at Illinois State University, July 7-11, 2008 Vertex Operator Algebras and Related Areas Conference in Honor of Geoffrey Mason’s 60th Birthday

– p. Introduction

Presented at Illinois State University, July 7-11, 2008 Vertex Operator Algebras and Related Areas Conference in Honor of Geoffrey Mason’s 60th Birthday

– p. Introduction

Presented at Illinois State University, July 7-11, 2008 Vertex Operator Algebras and Related Areas Conference in Honor of Geoffrey Mason’s 60th Birthday

“The mathematical universe is inhabited not only by important species but also by interesting individuals” – C. L. Siegel

– p. Introduction (I)

In 1983 Feingold and Frenkel studied the rank 3 hyperbolic Kac–Moody algebra = A++ with F 1 y y @ y Dynkin diagram @ 1 0 1 − Weyl Group W ( )= w , w , w wI = 2, wI wJ = mIJ F h 1 2 3 || | | | i 2 1 0 − 2(αI , αJ ) Cartan matrix [AIJ ]= 1 2 2 =  − −  (αJ , αJ ) 0 2 2  −    2 3 2 Coxeter exponents [mIJ ]= 3 2  ∞  2 2  ∞    – p. Introduction (II)

The simple roots, α 1, α0, α1, span the − 1 Root lattice Λ( )= α = nI αI n 1,n0,n1 Z F − ∈ ( I= 1 ) X−

– p. Introduction (II)

The simple roots, α 1, α0, α1, span the − 1 Root lattice Λ( )= α = nI αI n 1,n0,n1 Z F − ∈ ( I= 1 ) X−

which may be given as the space of integral symmetric (2 2)-matrices × + x z + Λ( )= X = x ,x−,z Z F z x ∈ ( " − # )

with Lorentzian inner product

(αI , αJ ) = det(αI ) + det(αJ ) det(αI + αJ ) −

– p. Introduction (III)

The matrices giving the simple roots are:

1 0 1 1 0 1 α 1 = α0 = − − α1 = − 0 1 1 0 1 0 " − # " − # " #

Λ( ) is a lattice in the space H2(R) of real symmetric (2 F 2)-matrices with that Lorentzian inner product. ×

– p. Introduction (III)

The matrices giving the simple roots are:

1 0 1 1 0 1 α 1 = α0 = − − α1 = − 0 1 1 0 1 0 " − # " − # " #

Λ( ) is a lattice in the space H2(R) of real symmetric (2 F 2)-matrices with that Lorentzian inner product. × The Weyl group W ( ) of generated by fundamental reﬂections F F 2(α, αI ) wI (α)= α αI − (αI , αI ) is a discrete subgroup of O(1, 2; R).

– p. Introduction (IV)

On Λ( ) can write reﬂections as matrix conjugations F X X T wI ( )= MI MI with

0 1 1 1 1 0 M 1 = , M0 = − , M1 = − 1 0 1 0 0 1 " # " # " − #

– p. Introduction (IV)

On Λ( ) can write reﬂections as matrix conjugations F X X T wI ( )= MI MI with

0 1 1 1 1 0 M 1 = , M0 = − , M1 = − 1 0 1 0 0 1 " # " # " − #

Then Feingold–Frenkel got the isomorphism

W ( ) = P GL (Z) F ∼ 2

Projective due to quadratic appearance of MI .

– p. Introduction (V)

The even part of the Weyl group

W +( ) = P SL (Z) F ∼ 2 is generated by

0 1 1 1 S = M 1M0 = , T = M0M1 = − − − 1 0 0 1 " − # " − #

– p. Introduction (V)

The even part of the Weyl group

W +( ) = P SL (Z) F ∼ 2 is generated by

0 1 1 1 S = M 1M0 = , T = M0M1 = − − − 1 0 0 1 " − # " − #

= Modular group P SL (Z) in hyperbolic Weyl group! ⇒ 2 Nice description of W +( ). F [[FF] also discussed relation to Siegel modular forms.]

– p. Introduction (VI)

P GL2(Z) acting on H2(R) preserves each hyperboloid (X, X)= c, giving model of hyperbolic space, Poincaré Disk:

– p. Introduction (VI)

P GL2(Z) acting on H2(R) preserves each hyperboloid (X, X)= c, giving model of hyperbolic space, Poincaré Disk:

r0r1r0α−1 r1r0α−1 r0r1r0α1 r1r0α1

r0r1α0 r1α0 r0α1 α1

α0

r 0 α −1 α −1 r r r r α r r r r α 0 1 −1 0 1 −1 α −1 1 0 1 0

r−1r0α1

r0r−1r0r1r0α1 r−1r0r1r0α1

r−1r0r1α0 r−1r0r1r0α−1

– p. General Structure (I)

Over-extension process of ﬁnite-dimensional simple g proceeds via non-twisted afﬁne extension g+:

αi for i = 1,...,ℓ : simple roots of g θ highest root of g A2 y 2

y 1

– p. General Structure (I)

Over-extension process of ﬁnite-dimensional simple g proceeds via non-twisted afﬁne extension g+:

αi for i = 1,...,ℓ : simple roots of g

θ highest root of g + A2 y 2

y Deﬁne afﬁne simple root by 0@ @y 1 α = δ θ ; null root δ orthogonal to αi of g 0 −

– p. General Structure (I)

Over-extension process of ﬁnite-dimensional simple g proceeds via non-twisted afﬁne extension g+:

αi for i = 1,...,ℓ : simple roots of g

θ highest root of g ++ A2 y 2

y y Define affine simple root by 1 0@ − @y 1 α = δ θ ; null root δ orthogonal to αi of g 0 − Define hyperbolic simple root by 2(α 1, α0)= (α0, α0) − − same length as α0, orthogonal to αi of g Root lattice Λ(g++) always Lorentzian. g++ often hyperbolic.

– p. General Structure (II)

++ (2 2) symmetric matrix presentation of A1 root lattice ﬁts × T into a family of Hermitian matrices X = X† = X¯ over K

+ ++ x z + Λ(g )= X = x ,x− Z, z Q(g) H2(K) z¯ x ∈ ∈ ⊂ ( " − # )

where K is one of the four normed division algebras over R: K = R, C, H, O equipped with conjugation z z¯. → N(z) =zz ¯ = zz¯ , N(z1z2)= N(z1)N(z2) z z =z ¯ z¯ , z z = 0 z = 0 or z = 0 1 2 2 1 1 2 ⇒ 1 2 and N(z1 + z2) N(z1) N(z2)= z1z¯2 + z2z¯1 = (z1,z2) is a Euclidean inner− product− on K.

– p. 10 General Structure (II)

++ (2 2) symmetric matrix presentation of A1 root lattice ﬁts × T into a family of Hermitian matrices X = X† = X¯ over K

+ ++ x z + Λ(g )= X = x ,x− Z, z Q(g) H2(K) z¯ x ∈ ∈ ⊂ ( " − # )

where K is one of the four normed division algebras over R: K = R, C, H, O equipped with conjugation z z¯. → K = R real, commutative, associative dimR(K) = 1 K = C commutative, associative dimR(K) = 2 K = H associative dimR(K) = 4 K = O alternative dimR(K) = 8 = Hyperbolic KMAs of ranks 3, 4, 6, 10 ⇒

– p. 10 General Structure (III)

We have a Lorentzian inner product on H2(K)

(X , X ) = det(X ) + det(X ) det(X + X ) 1 2 1 2 − 1 2 which, restricted to the subspace of H2(K) with zero diagonal

0 z1 0 z2 X1 = , X2 = H2(K) " z¯1 0 # " z¯2 0 # ∈ gives the Euclidean inner product

(X1, X2)= z1z¯2 + z2z¯1 = (z1,z2) so that subspace is isometric to K.

– p. 11 General Structure (IV)

We can ﬁnd ai K for i = 1,...,ℓ s.t. g Cartan matrix comes from K Euclidean∈ inner product

(ai, aj)= aia¯j + aja¯i

and the ﬁnite root lattice Q(g) K is their integral span. ⊂ Always can choose lengths so that θθ¯ = 1.

– p. 12 General Structure (IV)

We can ﬁnd ai K for i = 1,...,ℓ s.t. g Cartan matrix comes from K Euclidean∈ inner product

(ai, aj)= aia¯j + aja¯i

and the ﬁnite root lattice Q(g) K is their integral span. ⊂ Always can choose lengths so that θθ¯ = 1. Prop 1. We get g++ Cartan matrix from simple roots

1 0 1 θ 0 ai α 1 = , α0 = − − , αi = − 0 1 θ¯ 0 a¯i 0 " − # " − # " # for i = 1,...,ℓ.

– p. 12 General Structure (V)

Thm 1. Fundamental Weyl reﬂections of W W (g++) are ≡

wI (X)= MI X¯M † , I = 1, 0, 1,...,ℓ I − with unit versions of g simple roots εi = ai/ N(ai) and p 0 1 θ 1 εi 0 M 1 = , M0 = − , Mi = − 1 0 0 θ¯ 0 ε¯i " # " # " − #

– p. 13 General Structure (V)

Thm 1. Fundamental Weyl reﬂections of W W (g++) are ≡

wI (X)= MI X¯M † , I = 1, 0, 1,...,ℓ I −

with unit versions of g simple roots εi = ai/ N(ai) and p 0 1 θ 1 εi 0 M 1 = , M0 = − , Mi = − 1 0 0 θ¯ 0 ε¯i " # " # " − # Remarks Formula well-deﬁned for all K, including octonions Involves complex conjugation of X, not seen for F εi = ai only if g not simply laced 6 – p. 13 General Structure (VI)

– p. 14 General Structure (VI)

For generalizations of modular group P SL2(Z) need Thm 2. Even Weyl group W + W +(g++) is generated by ≡ X X (w 1wi)( )= Si Si† , i = 0, 1,...,ℓ − with 0 θ 0 εi S0 = , Si = − , i = 1,...,ℓ θ¯ 1 ε¯i 0 " − # " # Remarks Formula well-deﬁned for all K, including octonions W + SO(1,ℓ + 1; R). If det were deﬁned: det S = 1 ⊂ Does not involve complex conjugation of X = matrix subgroups of P SL (K) in associative cases! ⇒ 2 – p. 14 ++ Example: A (I) ++ 2 A2 y 2

y y 1 0@ − @y 1

– p. 15 ++ Example: A (I) ++ 2 A2 y 2

y y 1 0@ − @y 1 A2 is simply laced so εi = ai. Choose

3 1+ √3 i a2 √ t t θ a1 = 1 , a2 = 1= − JJ]JJ 2 J θ = a1 + a2 = a¯2 t Jr -ta ⇒ − J 1 J t J t

– p. 15 ++ Example: A (I) ++ 2 A2 y 2

y y 1 0@ − @y 1 A2 is simply laced so εi = ai. Choose

3 1+ √3 i a2 √ t t θ a1 = 1 , a2 = 1= − JJ]JJ 2 J θ = a1 + a2 = a¯2 t Jr -ta ⇒ − J 1 J t J t Finite root lattice

Q(A )= ma + na m,n Z C 2 { 1 2 | ∈ } ⊂ is the lattice (ring) of Eisenstein integers E.

Q(A2) densely packed ! E satisﬁes Euclidean algorithm

– p. 15 ++ Example: A2 (II)

+ ++ 1+√3 i Generating matrices of W (A2 ) (θ = 2 )

0 θ 0 1 0 θ¯ S0 = , S1 = − , S2 = θ¯ 1 1 0 θ 0 " − # " # " − #

– p. 16 ++ Example: A2 (II)

+ ++ 1+√3 i Generating matrices of W (A2 ) (θ = 2 )

0 θ 0 1 0 θ¯ S0 = , S1 = − , S2 = θ¯ 1 1 0 θ 0 " − # " # " − #

From these (θ3 = 1) −

θ 0 2 1 1 S1S2 = ¯ , S2S0(S1S2) = " 0 θ # " 0 1 # which together with S1 generate

a b SL2(E)= X = det(X) = 1, a, b, c, d E c d ∈ ( " # )

– p. 16

++ Example: A2 (III)

+ ++ E Prop 2. W (A2 ) ∼= P SL2( ).

– p. 17 ++ Example: A2 (III)

+ ++ E Prop 2. W (A2 ) ∼= P SL2( ). Outline of proof:

1 a 1 0 + ++ Show that , W (A2 ) " 0 1 # " a 1 # ∈ for all a E, i.e. can be generated from S , S , S ∈ 0 1 2 Take A SL (E) arbitrary. Use Euclidean algorithm to ∈ 2 bring A to diagonal form. Since det A = 1 it has to be of the form θm 0 = (S S )m ¯m 1 2 " 0 θ # for some m = 0,..., 5. Hence A W +(A++). ∈ 2

– p. 17 ++ Example: D4 (I) w 2

w w w w 101 3 − w 4

– p. 18 ++ Example: D4 (I) w 2

w w w w 101 3 − w 4 In the quaternions (non-commutative) two rings of integers are relevant:

Lipschitz: L = n + n i + n j + n k ni Z { 0 1 2 3 | ∈ } 1 Hurwitz: H = n + n i + n j + n k ni (Z + ) L { 0 1 2 3 | ∈ 2 }∪

– p. 18 ++ Example: D4 (I) w 2

w w w w 101 3 − w 4 In the quaternions (non-commutative) two rings of integers are relevant:

Lipschitz: L = n + n i + n j + n k ni Z { 0 1 2 3 | ∈ } 1 Hurwitz: H = n + n i + n j + n k ni (Z + ) L { 0 1 2 3 | ∈ 2 }∪

We ﬁnd simple roots of D4 (simply laced so εi = ai): 1 a = 1 a = 1+ i j k 1 2 2 − − − 1 1 a = 1 i + j k a = 1 i j + k 3 2 − − − 4 2 − − − Highest root: θ = 2a + a + a + a = 1 1 i j k 1 2 3 4 2 − − −

– p. 18 ++ Example: D4 (II)

What is the quaternionic matrix group generated by

0 θ 0 εi S0 = , Si = − , i = 1, ,ℓ ? θ¯ 1 ε¯i 0 ··· " − # " #

– p. 19 ++ Example: D4 (II)

What is the quaternionic matrix group generated by

0 θ 0 εi S0 = , Si = − , i = 1, ,ℓ ? θ¯ 1 ε¯i 0 ··· " − # " #

Usual determinant no longer deﬁned. But: SS† is Hermitian for any S; det(SS†) = det(S†S). Deﬁne

a b SL2(H)= S = det(SS†) = 1, a, b, c, d H c d ∈ ( " # )

Subgroup of all invertible (2 2) Hurwitz matrices. P SL (H) is quotient by normal× subgroup 11 . 2 {± }

– p. 19 ++ Example: D4 (II)

What is the quaternionic matrix group generated by

0 θ 0 εi S0 = , Si = − , i = 1, ,ℓ ? θ¯ 1 ε¯i 0 ··· " − # " #

Usual determinant no longer deﬁned. But: SS† is Hermitian for any S; det(SS†) = det(S†S). Deﬁne

a b SL2(H)= S = det(SS†) = 1, a, b, c, d H c d ∈ ( " # )

Subgroup of all invertible (2 2) Hurwitz matrices. P SL (H) is quotient by normal× subgroup 11 . 2 {± } + ++ H W (D4 ) is a proper subgroup of P SL2( ).

– p. 19 ++ Example: D4 (III)

+ Diagonal matrices (ﬁnite W (D4)) are always

a 0 S = with units a, d H and ad L " 0 d # ∈ ∈

– p. 20 ++ Example: D4 (III)

+ Diagonal matrices (ﬁnite W (D4)) are always

a 0 S = with units a, d H and ad L " 0 d # ∈ ∈

To extend to arbitrary matrices we need Deﬁnition. Let C = H[H, H]H be the two-sided ideal of H generated by all commutators [a, b] for a, b H. ∈ Find that C L H with index [L : C] = 2, and the quotient ⊂ ⊂ ring H/C is the ﬁeld of four elements F4. Also, nonzero elements of C have length at least 2, so there are no units of H in C. Then for a, d H units, ad L means ad 1 (mod C). ∈ ∈ ≡

– p. 20 ++ Example: D4 (IV)

Deﬁnition. The composition of the group homomorphism

P SL (H) P SL (H/C) 2 → 2 with the usual determinant of a (2 2)-matrix over the ﬁeld H C F × / ∼= 4, deﬁnes a group homomorphism

Det : P SL (H) F∗ 2 → 4 F Z onto 4∗ ∼= 3, the cyclic group of order 3.

– p. 21 ++ Example: D4 (IV)

Deﬁnition. The composition of the group homomorphism

P SL (H) P SL (H/C) 2 → 2 with the usual determinant of a (2 2)-matrix over the ﬁeld H C F × / ∼= 4, deﬁnes a group homomorphism

Det : P SL (H) F∗ 2 → 4 F Z onto 4∗ ∼= 3, the cyclic group of order 3. Deﬁnition. The kernel of Det is the subgroup

P SL(0)(H)= S P SL (H) Det(S) = 1 2 ∈ 2

– p. 21 ++ Example: D4 (V)

+ ++ (0) H Prop 3. W (D4 ) ∼= P SL2 ( )

– p. 22 ++ Example: D4 (V)

+ ++ (0) H Prop 3. W (D4 ) ∼= P SL2 ( ) Outline of proof:

Establish generating set of SL2(H), similar to SL2(E) construction [Krieg 1985].

– p. 22 ++ Example: D4 (V)

+ ++ (0) H Prop 3. W (D4 ) ∼= P SL2 ( ) Outline of proof:

Establish generating set of SL2(H), similar to SL2(E) construction [Krieg 1985]. (0) H Show that the following elements generate P SL2 ( ): for all units a, b H where ab 1 C ∈ − ∈ 1 1 0 1 a 0 , , " 0 1 # " 1 0 # " 0 b #

– p. 22 ++ Example: D4 (V)

+ ++ (0) H Prop 3. W (D4 ) ∼= P SL2 ( ) Outline of proof:

+ ++ Show that products of generators of W (D4 ) give (0) H those generators of P SL2 ( ).

– p. 22

Example: E++ (I) 8 y8

y y yyyyyyy 101234567 −

– p. 23 Example: E++ (I) 8 y8

y y yyyyyyy 101234567 −

O has basis 1,e , ,e with 1 ··· 7 1 2 e = 1= eiei ei , i = 1, , 7. i − +1 +3 ··· The 7 associative triples ↔ 7 lines in Fano Plane 4 6 ↔ H H 7 7 i = 1,ei,ei+1,ei+3 = . ∼ Hc h i Let i = ej j = i, i + 1, i + 3 . Hch | 6 i H If ej0 exchange it with 1 i to i ∈ ∈ Hj0 Hc,j0 2 3 5 get i and i . H Hc Hj0 Hc,j0 Let i, i , i , i be “Hurwitz" elts Z Z 1 (coefﬁcients all in or all in + 2 .

– p. 23 ++ Example: E8 (II)

H Hc Hj0 Hc,j0 Then all elements i + i and i + i form the (non-assoc.) ring of octavians O [Conway, Smith 2003].

– p. 24 ++ Example: E8 (II)

H Hc Hj0 Hc,j0 Then all elements i + i and i + i form the (non-assoc.) ring of octavians O [Conway, Smith 2003]. In O we can ﬁnd simple roots of E8 [Coxeter 1946]: 1 a = e a = e e e + e 1 3 2 2 − 1 − 2 − 3 4 1 a = e a = 1 e e + e 3 1 4 2 − − 1 − 4 5 1 a = 1 a = 1 e e e 5 6 2 − − 5 − 6 − 7 1 a = e a = 1+ e + e + e . 7 6 8 2 − 2 4 7

– p. 24 ++ Example: E8 (II)

– p. 24 ++ Example: E8 (III)

Remarks:

Finite W +(E ) SO(8; R). [Sudbery 1984] 8 ⊂ R O O so(8; ) ∼= Im( )+ G2 + Im( ) 8 P SO(8; R) on O = R : z (aLγ(z))aR ( ) ⇒ ∼ 7→ ∗ with N(aL)= N(aR) = 1 and γ Aut(O) = G . ∈ ∼ 2

– p. 25 ++ Example: E8 (III)

Remarks:

Finite W +(E ) SO(8; R). [Sudbery 1984] 8 ⊂ R O O so(8; ) ∼= Im( )+ G2 + Im( ) 8 P SO(8; R) on O = R : z (aLγ(z))aR ( ) ⇒ ∼ 7→ ∗ with N(aL)= N(aR) = 1 and γ Aut(O) = G . ∈ ∼ 2 + + . Z Discrete version: W (E8) ∼= O8 (2) 2 and Aut(O)= G2(2) with order 12096

240 Aut(O) 240 = W (E ) ×| | × | 8 |

– p. 25 ++ Example: E8 (III)

Remarks:

240 Aut(O) 240 = W (E ) ×| | × | 8 | Formula (*) not correct for ﬁnite transformations (non-associativity). No closed formula known...

– p. 25 ++ Example: E8 (IV)

But: Theorems 1 and 2 still apply so only have iterative + + ++ descriptions of W (E8) and W (E8 ) as groups generated by reﬂections.

– p. 26 ++ Example: E8 (IV)

But: Theorems 1 and 2 still apply so only have iterative + + ++ descriptions of W (E8) and W (E8 ) as groups generated by reﬂections.

For hyperbolic extension can only say

+ ++ O W (E8 ) ∼= P SL2( ) where right side is so far only understood as the group generated by all iterations of the conjugations

X SiXS†. Open problem to ﬁnd intrinsic deﬁnition of → i P SL2(O) in terms of matrices and automorphisms.

– p. 26 More results

K g Ring W (g) W +(g++) R A Z 2 Z P SL (Z) 1 ≡ 2 2 C A2 Eisenstein E Z3 ⋊ 2 P SL2(E) C B C Gaussian G Z ⋊ 2 P SL (G) ⋊ 2 2 ≡ 2 4 2 C G2 Eisenstein E Z6 ⋊ 2 P SL2(E) ⋊ 2 H I (0) I A4 Icosians S5 P SL2 ( ) H R 4 ⋊ (0) H ⋊ B4 Octahedral 2 S4 P SL2 ( ) 2 (0) H R 4 ⋊ ] H ⋊ C4 Octahedral 2 S4 P SL2 ( ) 2 H H 3 ⋊ (0) H D4 Hurwitz 2 S4 P SL2 ( ) H F Octahedral R 25 ⋊ (S S ) P SL (H) ⋊ 2 4 3 × 3 2 O O 7 ⋊ (0) O D8 Octavians 2 S8 P SL2 ( ) O O 8 ⋊ (0) O ⋊ B8 Octavians 2 S8 P SL2 ( ) 2 + O E8 Octavians O 2 . O (2) . 2 P SL2(O) 8 – p. 27 Relations to physics

– p. 28 Relations to physics