Block Decomposition of a Class of Integrable Representations of - PowerPoint PPT Presentation

Block Decomposition of a Class of Integrable Representations of Toroidal Lie Algebras Tanusree Khandai Indian Institute of Science Education and Research, Mohali Interactions of quantum affine algebras with cluster algebras, current algebras and categorification - Conference celebrating 60th birthday of Vyjayanthi Chari

Notations Let g a complex finite-dimensional simple Lie algebra ; h a Cartan subalgebra of g . { α i : 1 ≤ i ≤ n } := simple roots of g , { ω 1 , · · · , ω n } := fundamental weights of g , Q fin = � n i = 1 Z α i the root lattice, P fin = � n i = 1 Z ω i , weight lattice and P + fin = � n i = 1 Z + ω i dominant integral weights of g θ the highest root of g and θ ∨ the corresponding co-root; - g aff affine Kac-Moody algebra associated with g h aff a Cartan subalgebra of g aff ( . | . ) the Killing form on g ; { α i : 1 ≤ 0 ≤ n } := simple roots of g aff { Λ 1 , · · · , Λ n , Λ 0 } := fundamental weights of g aff Q aff = � n i = 0 Z α i the root lattice P aff = � n i = 0 Z Λ i , weight lattice and P + aff = � n i = 0 Z + Λ i dominant integral weights of g aff

Notations Let g a complex finite-dimensional simple Lie algebra ; h a Cartan subalgebra of g . { α i : 1 ≤ i ≤ n } := simple roots of g , { ω 1 , · · · , ω n } := fundamental weights of g , Q fin = � n i = 1 Z α i the root lattice, P fin = � n i = 1 Z ω i , weight lattice and P + fin = � n i = 1 Z + ω i dominant integral weights of g θ the highest root of g and θ ∨ the corresponding co-root; - g aff affine Kac-Moody algebra associated with g h aff a Cartan subalgebra of g aff ( . | . ) the Killing form on g ; { α i : 1 ≤ 0 ≤ n } := simple roots of g aff { Λ 1 , · · · , Λ n , Λ 0 } := fundamental weights of g aff Q aff = � n i = 0 Z α i the root lattice P aff = � n i = 0 Z Λ i , weight lattice and P + aff = � n i = 0 Z + Λ i dominant integral weights of g aff

Toroidal Lie Algebra Definition A k -toroidal Lie algebra associated with g is a Lie algebra with underlying vector space T k ( g ) := g ⊗ C [ t ± 1 1 , · · · , t ± k k ] ⊕ D k ⊕ Z , where, D k is the space spanned by k derivations d 1 , · · · , d k , Z is an infinite-dimensional space spanned by Z k -graded central elements { t m c i , m ∈ Z k , 1 ≤ i ≤ k } , together with the relation � k i = 1 r i t r c i = 0 ; and Lie bracket : k [ x ⊗ t m , y ⊗ t s ] = [ x , y ] ⊗ t m + s + � m i t m + s c i ( x | y ) , i = 1 d i ( x ⊗ t m ) = m i x ⊗ t m , ∀ x ∈ g . Let h tor := h ⊕ D k ⊕ C c 1 ⊕ · · · ⊕ C c k where c 1 , · · · , c k are the zero graded central elements in T k ( g ) .

Toroidal Lie Algebra Definition A k -toroidal Lie algebra associated with g is a Lie algebra with underlying vector space T k ( g ) := g ⊗ C [ t ± 1 1 , · · · , t ± k k ] ⊕ D k ⊕ Z , where, D k is the space spanned by k derivations d 1 , · · · , d k , Z is an infinite-dimensional space spanned by Z k -graded central elements { t m c i , m ∈ Z k , 1 ≤ i ≤ k } , together with the relation � k i = 1 r i t r c i = 0 ; and Lie bracket : k [ x ⊗ t m , y ⊗ t s ] = [ x , y ] ⊗ t m + s + � m i t m + s c i ( x | y ) , i = 1 d i ( x ⊗ t m ) = m i x ⊗ t m , ∀ x ∈ g . Let h tor := h ⊕ D k ⊕ C c 1 ⊕ · · · ⊕ C c k where c 1 , · · · , c k are the zero graded central elements in T k ( g ) .

Integrable Representation of Lie algebra Definition A T k ( g ) -module V is said integrable if V = ⊕ V µ , where V µ = { v ∈ V : h . v = µ ( h ) v , for all h ∈ h } . µ ∈ h ∗ tor the root vectors corresponding to the real roots of T k ( g ) act nilpotently on every non-zero vector of V . For an integer k , let I k fin be the category of integral T k ( g ) -modules with finite-dimensional weight spaces.

Irreducible objects of I 1 fin For k = 1 , T k ( g ) = g aff (untwisted affine Kac-Moody algebra associated with g ): V. Chari, A. Pressley The simple objects of I 1 fin on which the center acts trivially are of the form → → b , s ) := V b 1 ( λ 1 ) ⊗ · · · ⊗ V b r ( λ r ) ⊗ t s C [ t ± m V ( λ, ] 1 → → fin ) r , b = ( b 1 , · · · , b r ) ∈ ( C × ) r for r ∈ Z + , λ = ( λ 1 , · · · , λ r ) ∈ ( P + where, m ∈ Z + and 0 ≤ s ≤ m − 1 , and the simple objects of I 1 fin on which the center acts non-trivially are : standard modules of the form X (Λ) , with Λ ∈ P + aff or restricted duals of standard modules. Here, for λ ∈ P + , b ∈ C × , V b ( λ ) is the evaluation module for the loop algebra g ⊗ C [ t ± 1 1 ] , with underlying vector space V ( λ ) , and g ⊗ C [ t ± 1 1 ] action: x ⊗ t r 1 . v = b r x . v , ∀ x ∈ g , v ∈ V ( λ ) .

Irreducible objects of I 1 fin For k = 1 , T k ( g ) = g aff (untwisted affine Kac-Moody algebra associated with g ): V. Chari, A. Pressley The simple objects of I 1 fin on which the center acts trivially are of the form → → b , s ) := V b 1 ( λ 1 ) ⊗ · · · ⊗ V b r ( λ r ) ⊗ t s C [ t ± m V ( λ, ] 1 → → fin ) r , b = ( b 1 , · · · , b r ) ∈ ( C × ) r for r ∈ Z + , λ = ( λ 1 , · · · , λ r ) ∈ ( P + where, m ∈ Z + and 0 ≤ s ≤ m − 1 , and the simple objects of I 1 fin on which the center acts non-trivially are : standard modules of the form X (Λ) , with Λ ∈ P + aff or restricted duals of standard modules. Here, for λ ∈ P + , b ∈ C × , V b ( λ ) is the evaluation module for the loop algebra g ⊗ C [ t ± 1 1 ] , with underlying vector space V ( λ ) , and g ⊗ C [ t ± 1 1 ] action: x ⊗ t r 1 . v = b r x . v , ∀ x ∈ g , v ∈ V ( λ ) .

Irreducible objects of I 1 fin For k = 1 , T k ( g ) = g aff (untwisted affine Kac-Moody algebra associated with g ): V. Chari, A. Pressley The simple objects of I 1 fin on which the center acts trivially are of the form → → b , s ) := V b 1 ( λ 1 ) ⊗ · · · ⊗ V b r ( λ r ) ⊗ t s C [ t ± m V ( λ, ] 1 → → fin ) r , b = ( b 1 , · · · , b r ) ∈ ( C × ) r for r ∈ Z + , λ = ( λ 1 , · · · , λ r ) ∈ ( P + where, m ∈ Z + and 0 ≤ s ≤ m − 1 , and the simple objects of I 1 fin on which the center acts non-trivially are : standard modules of the form X (Λ) , with Λ ∈ P + aff or restricted duals of standard modules. Here, for λ ∈ P + , b ∈ C × , V b ( λ ) is the evaluation module for the loop algebra g ⊗ C [ t ± 1 1 ] , with underlying vector space V ( λ ) , and g ⊗ C [ t ± 1 1 ] action: x ⊗ t r 1 . v = b r x . v , ∀ x ∈ g , v ∈ V ( λ ) .

Block Decomposition of the category I 1 fin [ 0 ] Let I 1 fin [ 0 ] be the subcategory of level zero objects of g aff . V. Chari, J. Greenstein, A. Moura The objects of I 1 fin [ 0 ] can be written as direct sum of indecomposables.

Block Decomposition of the category I 1 fin [ 0 ] Let I 1 fin [ 0 ] be the subcategory of level zero objects of g aff . V. Chari, J. Greenstein, A. Moura The objects of I 1 fin [ 0 ] can be written as direct sum of indecomposables. An indecomposable object in I 1 fin [ 0 ] has finitely many simple constituents which are non-trivial as modules over g ⊗ C [ t 1 , t − 1 1 ] .

Block Decomposition of the category I 1 fin [ 0 ] Let I 1 fin [ 0 ] be the subcategory of level zero objects of g aff . V. Chari, J. Greenstein, A. Moura The objects of I 1 fin [ 0 ] can be written as direct sum of indecomposables. An indecomposable object in I 1 fin [ 0 ] has finitely many simple constituents which are non-trivial as modules over g ⊗ C [ t 1 , t − 1 1 ] . Given two irreducible g aff -modules, → → → → V ( λ, b , s ) and V ( µ, a , p ) , → → fin ) r and → → µ ∈ ( P + a ∈ ( C × ) r , there exists a sequence of with λ, b , indecomposable g aff -modules V 1 , V 2 , · · · , V r in I 1 fin [ 0 ] such that Hom ( V i , V i + 1 ) � = 0 , or Hom ( V i + 1 , V i ) � = 0 , if and only if → → a for some s ∈ C ∗ and b = s . → → λ − µ ∈ ( Q fin ) r .

Block Decomposition of the category I 1 fin [ 0 ] Let I 1 fin [ 0 ] be the subcategory of level zero objects of g aff . V. Chari, J. Greenstein, A. Moura The objects of I 1 fin [ 0 ] can be written as direct sum of indecomposables. An indecomposable object in I 1 fin [ 0 ] has finitely many simple constituents which are non-trivial as modules over g ⊗ C [ t 1 , t − 1 1 ] . The category I 1 fin [ 0 ] decomposes into blocks which are parametrized by orbits for the natural action of the group C × on Ξ := { f : C × → P fin / Q fin : f ( a ) = 0 for all but finitely many a ∈ C × } . Equivalence classes of indecomposable objects in Blocks in I 1 fin [ 0 ] ← → I 1 fin [ 0 ] with respect to the equivalence relation ∼ defined as follows. X ∼ Y if there exists a sequence of indecomposable modules X = X 1 , · · · , X k = Y in I 1 fin [ 0 ] such that for 1 ≤ i ≤ k − 1 , Hom ( X i , X i + 1 ) � = 0 , or Hom ( X i + 1 , X i ) � = 0

Irreducible Modules in I k fin for k > 1 S. E Rao, 2004 For k ≥ 2 , the simple objects in I k fin on which the central elements act trivially are of the form, → → → → a , s ) := V a 1 ( λ 1 ) ⊗ · · · ⊗ V a r ( λ r ) ⊗ t s C [ t G ( a ) ] λ , V ( λ, → → fin ) r , a = ( a 1 , · · · , a r ) ∈ (( C × ) k ) r for r ∈ Z + , λ = ( λ 1 , · · · , λ r ) ∈ ( P + where, → → a ) is a subgroup of Z k of rank k and t s ∈ C [ t ± 1 , · · · , t ± 1 G ( ] . λ, 1 k V a ( λ ) is the evaluation module for g ⊗ C [ t ± 1 , · · · , t ± 1 ] , with underlying vector 1 k space V ( λ ) , with λ ∈ P + fin , a ∈ ( C × ) k .