Classification of integrable modules of twisted full toroidal Lie - PowerPoint PPT Presentation

Classification of integrable modules of twisted full toroidal Lie algebras Punita Batra Harish-Chandra Research Institute Allahabad, INDIA 5 th June 2018 5 th June 2018 Punita Batra Classification of integrable modules of twisted full toroidal Lie algebras 1 / 32

Preliminaries All vector spaces, algebras and tensor products are over complex numbers C . Let Z , N and Z + denote integers, non-negative integers and positive integers. Let g be a finite dimensional simple Lie algebra and let ( , ) be a non-degenerate symmetric bilinear form on g . We fix a positive integer n . Let σ 0 , σ 1 , · · · , σ n be commuting finite order automorphisms of g of order m 0 , m 1 , · · · , m n respectively. Let m = ( m 1 , · · · , m n ) ∈ Z n . Let k = ( k 1 , · · · , k n ) and l = ( l 1 , · · · , l n ) denote vectors in Z n . 5 th June 2018 Punita Batra Classification of integrable modules of twisted full toroidal Lie algebras 2 / 32

Let Γ = m 1 Z ⊕ · · · ⊕ m n Z and Γ 0 = m 0 Z . Let Λ = Z n / Γ and Λ 0 = Z / Γ 0 . Let k and l denote the images in Λ. For any integers k 0 and l 0 , let k 0 and l 0 denote the images in Λ 0 . Let = C [ t ± 1 0 , · · · , t ± 1 A n ] , = C [ t ± 1 1 , · · · , t ± 1 A n n ] , = C [ t ± m 1 , · · · , t ± m n A ( m ) ] , 1 n = C [ t ± m 0 , t ± m 1 , · · · , t ± m n A ( m 0 , m ) ] . 0 1 n 5 th June 2018 Punita Batra Classification of integrable modules of twisted full toroidal Lie algebras 3 / 32

For k ∈ Z n , let t k = t k 1 1 · · · t k n n ∈ A n . Let Ω A be the vector space spanned by symbols t k 0 0 t k K i , 0 ≤ i ≤ n , k 0 ∈ Z , k ∈ Z n . Let dA be the subspace n � k i t k 0 0 t k K i . Let L ( g ) = g ⊗ A and define toroidal Lie algebra spanned by i =0 ∼ L ( g ) = L ( g ) ⊕ Ω A / dA . 0 t k and Y = Y ⊗ t l 0 0 t l for X , Y ∈ g , k 0 , l 0 ∈ Z and Let X ( k 0 , k ) = X ⊗ t k 0 k , l ∈ Z n . [ X ( k 0 , k ) , Y ( l 0 , l )] = [ X , Y ]( l 0 + k 0 , l + k ) + ( X , Y ) � k i t l 0 + k 0 t l + k K i . 0 Ω A / dA is central. 5 th June 2018 Punita Batra Classification of integrable modules of twisted full toroidal Lie algebras 4 / 32

We will now define multiloop algebra as a subalgebra of L ( g ). For 0 ≤ i ≤ n , let ξ i be a m i th primitive root of unity. Let g ( k 0 , k ) = { x ∈ g | σ i x = ξ k i i x , 0 ≤ i ≤ n } . Then define � g ( k 0 , k ) ⊗ t k 0 0 t k , L ( g , σ ) = ( k o , k ) ∈ Z n +1 which is called a multiloop algebra. The finite dimensional irreducible modules for L ( g , σ ) are classified by Michael Lau. 5 th June 2018 Punita Batra Classification of integrable modules of twisted full toroidal Lie algebras 5 / 32

We will now define the universal central extension of L ( g , σ ) . Define Ω A ( m 0 , m ) and dA ( m 0 , m ) similar to the definition of Ω A and dA by replacing A by A ( m 0 , m ). Denote Z ( m 0 , m ) = Ω A ( m 0 , m ) / dA ( m 0 , m ) and note that Z ( m 0 , m ) ⊆ Ω A / dA . Define ∼ L ( g , σ ) = L ( g , σ ) ⊕ Z ( m 0 , m ) . 0 t k and Let X ∈ g ( k 0 , k ) and Y ∈ g ( l 0 , l ) and let X ( k 0 , k ) = X ⊗ t k 0 Y ( l 0 , l ) = Y ⊗ t l 0 0 t l . Define (a) [ X ( k 0 , k ) , Y ( l 0 , l )] = [ X , Y ]( k 0 + l 0 , k + l ) + ( X , Y ) � k i t l 0 + k 0 t l + k K i . 0 (b) Z ( m 0 , m ) is central. 5 th June 2018 Punita Batra Classification of integrable modules of twisted full toroidal Lie algebras 6 / 32

Derivation algebra of A ( m 0 , m ) and its extension to Z ( m 0 , m ). Let D ( m 0 , m ) be the derivation algebra of A ( m 0 , m ). From now onwards we let s and r to be in Γ and s 0 and r 0 to be in Γ 0 . For 0 ≤ i ≤ n , consider t s 0 0 t s t i d dt i which acts on A ( m 0 , m ) as derivations. It is well known that D ( m 0 , m ) has the following basis d { t s 0 0 t s t i | 0 ≤ i ≤ n , s o ∈ Γ 0 , s ∈ Γ } . dt i Let d i = t i d dt i and it is easy to see that [ t s 0 0 t s d a , t r 0 0 t r d b ] = r a t r 0 + s 0 t r + s d b − s b t r 0 + s 0 t r + s d a . 0 0 5 th June 2018 Punita Batra Classification of integrable modules of twisted full toroidal Lie algebras 7 / 32

D ( m 0 , m ) acts on Z ( m 0 , m ) in the following way n 0 t r K b ) = r a t r 0 + s 0 � s p t r 0 + s 0 t s 0 0 t s d a . ( t r 0 t r + s K b + δ ab t r + s K p . It is known 0 0 p =0 that D ( m 0 , m ) admits two non-trivial 2-cocycles with values in Z ( m 0 , m ). n � r p t r 0 + s 0 ϕ 1 ( t r 0 0 t r d a , t s 0 t r + s K p , 0 t s d b ) = − s a r b 0 p =0 n � r p t r 0 + s 0 ϕ 2 ( t r 0 0 t r d a , t s 0 t r + s K p . 0 t s d b ) = r a s b 0 p =0 5 th June 2018 Punita Batra Classification of integrable modules of twisted full toroidal Lie algebras 8 / 32

Let ϕ be arbitrary linear combinations of ϕ 1 and ϕ 2 . Then there is a corresponding Lie algebra τ = L ( g , σ ) ⊕ Z ( m 0 , m ) ⊕ D ( m 0 , m ). The Lie brackets are defined in the following way. [ t r 0 0 t r d a , X ( k 0 , k )] = k a X ( k 0 + r 0 , k + r ) , n 0 t r d a , t s 0 t s K b ] = s a t r 0 + s 0 � r p t r 0 + s 0 [ t r 0 t r + s K b + δ ab t r + s K p , 0 0 p =0 [ t r 0 0 t r d a , t s 0 0 t s d b ] = s a t r 0 + s 0 t r + s d b − r b t r 0 + s 0 t r + s d a + ϕ ( t r 0 0 t r d a , t s 0 t s d b ), 0 0 where r , s ∈ Γ , r 0 , s 0 ∈ Γ 0 , X ∈ g ( k 0 , k ) . 5 th June 2018 Punita Batra Classification of integrable modules of twisted full toroidal Lie algebras 9 / 32

Assumptions (a) g (0 , 0) is simple Lie algebra. (b) We can choose Cartan subalgebra h (0) and h for g (0 , 0) and g such that h (0) ⊆ h . (c) It is known that ∆ × 0 = ∆( g (0 , 0) , h (0)) \{ 0 } is an irreducible reduced finite root system and has atmost two root lengths. Let ∆ × 0 , sh be the set of non-zero short roots. Define � ∆ × 0 ∪ 2∆ × 0 , sh if ∆ × 0 is of type B l ∆ × ∆ 0 , en = ∆ × 0 , en = 0 , en ∪ { 0 } . ∆ × otherwise 0 We assume that ∆( g , h (0)) = ∆ 0 , en . 5 th June 2018 Punita Batra Classification of integrable modules of twisted full toroidal Lie algebras 10 / 32

Root space decomposition and integrable modules for τ . First note the center of τ is spanned by K 0 , K 1 , · · · , K n . Let H = h (0) ⊕ � C K i ⊕ � C d i which is an abelian Lie algebra of τ and plays the role of Cartan subalgebra. Define δ i , w i ∈ H ∗ (0 ≤ i ≤ n ) be such that w i ( h (0)) = 0 , w i ( K j ) = δ ij , w i ( d j ) = 0 , δ i ( h (0)) = 0 , δ i ( K j ) = 0 , δ i ( d j ) = δ ij . 5 th June 2018 Punita Batra Classification of integrable modules of twisted full toroidal Lie algebras 11 / 32

n � k i δ i for k ∈ Z n . Let δ k = i =1 Let g ( k 0 , k , α ) = { x ∈ g ( k 0 , k ) | [ h , x ] = α ( h ) x for all h ∈ h (0) } then � τ has a root space decomposition. τ = τ β where β ∈ ∆ ∆ ⊆ { α + k 0 δ 0 + δ k , α ∈ ∆ 0 , e n , k 0 ∈ Z , k ∈ Z n } . 0 t k for α � = 0 , = g ( k 0 , k , α ) ⊗ t k 0 τ α + k 0 δ 0 + δ k n n 0 t k ⊕ = g ( k 0 , k , 0) ⊗ t k 0 � C t k 0 0 t k K i ⊕ � C t k 0 0 t k d i . τ k 0 δ 0 + δ k i =0 i =0 5 th June 2018 Punita Batra Classification of integrable modules of twisted full toroidal Lie algebras 12 / 32

Notice that τ 0 = H . Now we will define a non-degenerate bilinear form on H ∗ . For α ∈ h (0) ∗ extended α to H by α ( K i ) = α ( d i ) = 0 , 0 ≤ i ≤ n . Let ( h (0) , K i ) = 0 = ( h (0) , d i ), ( δ k + δ k 0 , δ l + δ l 0 ) = 0 = ( w i , w j ), ( δ i , w j ) = δ ij . The form on h (0) is the restriction of the form ( , ) on g . 5 th June 2018 Punita Batra Classification of integrable modules of twisted full toroidal Lie algebras 13 / 32

For γ = α + k 0 δ 0 + δ k is called real root if α � = 0 which is equivalent to ( γ, γ ) � = 0. Denote ∆ re be the set of real roots. For α ∈ ∆ 0 , en , denote α ∨ the co-root of α . n Define γ ∨ = α ∨ + � 2 k i K i for γ real. ( α,α ) i =0 Then γ ( γ ∨ ) = α ( α ∨ ) = 2. For γ real root, define reflection on H ∗ by r γ ( λ ) = λ − λ ( γ ∨ ) γ, λ ∈ H ∗ . Let W be the Weyl group genarated by r γ , γ ∈ ∆ re . 5 th June 2018 Punita Batra Classification of integrable modules of twisted full toroidal Lie algebras 14 / 32

Definition A module V of τ is called integrable if � V = V λ , V λ = { v ∈ V | hv = λ ( h ) v , h ∈ H } , dim V λ < ∞ , λ ∈ H ∗ 0 t k acts locally nilpotently on V for α � = 0. g ( k 0 , k , α ) ⊗ t k 0 Let P ( V ) = { γ ∈ H ∗ | V γ � = 0 } . For an irreducible integrable module with non zero central charge, we can assume that K 0 acts as C 0 > 0 and K i ( i � = 0) acts trivially upto a choice of co-ordinates. For any λ ∈ P ( V ) , λ ( K i ) = C i = 0 for 1 ≤ i ≤ n and λ ( K 0 ) = C 0 . Let α 0 = − β 0 + δ 0 where β 0 is maximal root in ∆ 0 , en . Note that α 0 may not be root of τ . Let α 1 , α 2 , · · · , α p be a set of simple roots for p ∆( g ( ◦ , ◦ ) , h (0)) and let Q + = � N α i . Define an ordering on H ∗ , λ ≤ µ i =0 for λ, µ ∈ H ∗ , if µ − λ ∈ Q + . 5 th June 2018 Punita Batra Classification of integrable modules of twisted full toroidal Lie algebras 15 / 32