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) 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 " − # " # – 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.