On the affine VW supercategory Mee Seong Im West Point, NY - PowerPoint PPT Presentation

On the affine VW supercategory On the affine VW supercategory Mee Seong Im West Point, NY Interactions of quantum affine algebras with cluster algebras, current algebras and categorification A conference celebrating the 60th birthday of Vyjayanthi Chari Catholic University of America, Washington, D.C. May 28, 2018 Mee Seong Im West Point, NY 1

On the affine VW supercategory Joint work Joint with Martina Balagovic, Zajj Daugherty, Inna Entova-Aizenbud, Iva Halacheva, Johanna Hennig, Gail Letzter, Emily Norton, Vera Serganova, and Catharina Stroppel. Mee Seong Im West Point, NY 2

On the affine VW supercategory Preliminaries Background: vector superspaces. Work over C . A Z / 2 Z -graded vector space V = V 0 ⊕ V 1 is a vector superspace . The superdimension of V is dim( V ) := (dim V 0 | dim V 1 ) = dim V 0 − dim V 1 . Given a homogeneous element v ∈ V , the parity (or the degree ) of v is v ∈ { 0 , 1 } . The parity switching functor π sends V 0 �→ V 1 and V 1 �→ V 0 . Let m = dim V 0 and n = dim V 1 . The Lie superalgebra is gl ( m | n ) := End C ( V ). That is, given a homogeneous ordered basis for V: V = C { v 1 , . . . , v m } ⊕ C { v 1 ′ , . . . , v n ′ } , � �� � � �� � V 0 V 1 Mee Seong Im West Point, NY 3

On the affine VW supercategory Preliminaries Matrix representation for gl ( m | n ). the Lie superalgebra is the endomorphism algebra �� A � � B : A ∈ M m , m , B , C t ∈ M m , n , D ∈ M n , n gl ( m | n ) := , C D where M i , j := M i , j ( C ). Since gl ( m | n ) = gl ( m | n ) 0 ⊕ gl ( m | n ) 1 , �� A �� �� 0 �� 0 B gl ( m | n ) 0 = and gl ( m | n ) 1 = . 0 D C 0 We say V is the natural representation of gl ( m | n ). The grading on gl ( m | n ) is induced by V , with Lie superbracket (supercommutator) [ x , y ] = xy − ( − 1) xy yx for x , y homogeneous. Mee Seong Im West Point, NY 4

On the affine VW supercategory Periplectic Lie superalgebras p ( n ) Periplectic Lie superalgebras p ( n ). Let m = n . Then V = C 2 n = C { v 1 , . . . , v n } ⊕ C { v 1 ′ , . . . , v n ′ } . � �� � � �� � V 0 V 1 Define β : V ⊗ V → C as a symmetric, odd, nondegenerate bilinear form satisfying: β ( v , w ) = β ( w , v ) , β ( v , w ) = 0 if v = w . We define periplectic (strange) Lie superalgebras as: p ( n ) := { x ∈ End C ( V ) : β ( xv , w ) + ( − 1) xv β ( v , xw ) = 0 } . In terms of above basis, �� A � � B ∈ gl ( n | n ) : B = B t , C = − C t p ( n ) = . − A t C Mee Seong Im West Point, NY 5

On the affine VW supercategory Periplectic Lie superalgebras p ( n ) Symmetric monoidal structure. Consider the category C of representations of p ( n ) with Hom p ( n ) ( V , V ′ ) := { f : V → V ′ : f homogeneous , C − linear , f ( x . v ) = ( − 1) xf x . f ( v ) , v ∈ V , x ∈ p ( n ) } . Then U ( p ( n )) of p ( n ) is a Hopf superalgebra: ◮ (coproduct) ∆( x ) = x ⊗ 1 + 1 ⊗ x , ◮ (counit) ǫ ( x ) = 0, ◮ (antipode) S(x) = -x. So the category of representations of p ( n ) is monoidal. For x ⊗ y ∈ U ( p ( n )) ⊗ U ( p ( n )) on v ⊗ w , ( x ⊗ y ) . ( v ⊗ w ) = ( − 1) yv xv ⊗ yw . Mee Seong Im West Point, NY 6

On the affine VW supercategory Periplectic Lie superalgebras p ( n ) Symmetric monoidal structure. For x , y , a , b ∈ U ( p ( n )), ( x ⊗ y ) ◦ ( a ⊗ b ) := ( − 1) ya ( x ◦ a ) ⊗ ( y ◦ b ) , and for two representations V and V ′ , the super swap σ : V ⊗ V ′ − → V ′ ⊗ V , σ ( v ⊗ w ) = ( − 1) vw w ⊗ v is a map of p ( n )-representations satisfying σ ∗ = − σ . Thus C is a symmetric monoidal category. Furthermore, β induces a representation V and its dual V ∗ via V → V ∗ , v �→ β ( v , − ) , identifying V 1 with V ∗ 0 and V 0 with V ∗ 1 . This induces the dual map � β ∗ : C ∼ = C ∗ − → ( V ⊗ V ) ∗ ∼ β ∗ (1) = = V ⊗ V , − v i ⊗ v i ′ + v i ′ ⊗ v i , i where β = β ∗ = 1. Mee Seong Im West Point, NY 7

On the affine VW supercategory Periplectic Lie superalgebras p ( n ) Quadratic (fake) Casimir and Jucys-Murphy elements: y ℓ ’s. Furthermore, we define � � � x ⊗ x ∗ ∈ p ( n ) ⊗ gl ( n | n ) Ω = 2 2Ω = − , x ∈X where X is a basis of p ( n ) and x ∗ is a dual basis element of p ( n ), and p ( n ) ⊥ is taken with respect to the supertrace: � A � B = tr ( A ) − tr ( D ) . str C D The actions of Ω and p ( n ) commute on M ⊗ V , so Ω ∈ End p ( n ) ( M ⊗ V ). We define ℓ − 1 � Y ℓ : M ⊗ V ⊗ a − → M ⊗ V ⊗ a as Y ℓ = Ω i ,ℓ = , i =0 where Ω i ,ℓ acts on the i -th and ℓ -th factor, and identity otherwise, where the 0-th factor is the module M . Mee Seong Im West Point, NY 8

On the affine VW supercategory Schur-Weyl duality Classical Schur-Weyl duality. Let W be an n -dimensional complex vector space. Consider W ⊗ a . Then the symmetric group S a acts on W ⊗ a by permuting the factors: for s i = ( i i + 1) ∈ S a , s i . ( w 1 ⊗ · · · ⊗ w a ) = w 1 ⊗ · · · ⊗ w i +1 ⊗ w i ⊗ · · · ⊗ w a . We also have GL ( W ) acting on W ⊗ a via the diagonal action: for g ∈ GL ( W ), g . ( w 1 ⊗ · · · ⊗ w a ) = gw 1 ⊗ · · · ⊗ gw a . Then actions of GL ( W ) (left natural action) and S a (right permutation action) commute giving us the following: Mee Seong Im West Point, NY 9

On the affine VW supercategory Schur-Weyl duality Schur-Weyl duality. Consider the natural representations φ ψ ( C S a ) op → End C ( W ⊗ a ) → End C ( W ⊗ a ) . − and GL ( W ) − Then Schur-Weyl duality gives us 1. φ ( C S a ) = End GL ( W ) ( W ⊗ a ), 2. if n ≥ a , then φ is injective. So im φ ∼ = End GL ( W ) ( W ⊗ a ), 3. ψ ( GL ( W )) = End C S a ( W ⊗ a ), 4. there is an irreducible ( GL ( W ) , ( C S a ) op )-bimodule decomposition (see next slide): Mee Seong Im West Point, NY 10

On the affine VW supercategory Schur-Weyl duality Schur-Weyl duality (continued). � W ⊗ a = ∆ λ ⊗ S λ , λ =( λ 1 ,λ 2 ,... ) ⊢ a ℓ ( λ ) ≤ n where ◮ ∆ λ is an irreducible GL ( W )-module associated to λ , ◮ S λ is an irreducible C S a -module associated to λ , and ◮ ℓ ( λ ) = max { i ∈ Z : λ i � = 0 , λ = ( λ 1 , λ 2 , . . . ) } . In higher Schur-Weyl duality, we construct a result analogous to C S a ∼ = End GL ( W ) ( W ⊗ a ) , but we use the existence of commuting actions on the tensor product of arbitrary gl n -representation M with W ⊗ a : gl n � M ⊗ W ⊗ a � H a , Mee Seong Im West Point, NY 11

On the affine VW supercategory Schur-Weyl duality where H a is the degenerate affine Hecke algebra . The Hecke algebra H a contains the group algebra C S a and the polynomial algebra C [ y 1 , . . . , y a ] as subalgebras. So as a vector space, H a ∼ = C S a ⊗ C [ y 1 , . . . , y a ], and has a basis B = { wy k 1 1 · · · y k a a : w ∈ S a , k i ∈ N 0 } . In this talk, we aim to construct higher Schur-Weyl duality in the context of p ( n ) and affine Brauer algebras, which we will denote by sV V a (so affine Brauer algebras were constructed from the motivation to formulate higher Schur-Weyl duality for the periplectic Lie superalgebra action, i.e., we need to find another algebra whose action on a representation M ⊗ V ⊗ a commutes with the action of p ( n )). Mee Seong Im West Point, NY 12

On the affine VW supercategory Affine Brauer algebras Affine Brauer algebras (generators and local moves). V a has generators s i , b i , b ∗ sV i , y j , where i = 1 , . . . , a − 1, j = 1 , . . . , a and relations = = = = = Continued in the next slide. Mee Seong Im West Point, NY 13

On the affine VW supercategory Affine Brauer algebras Affine Brauer algebras (local moves; continued). = − = − = (braid reln) = (braid reln) = (adjunctions) = − (adjunctions) = (untwisting reln) = = − (untwisting reln) = Mee Seong Im West Point, NY 14

On the affine VW supercategory Affine Brauer algebras Affine Brauer algebras (local moves; continued). = = = = = = = + − = − = − − = Mee Seong Im West Point, NY 15

On the affine VW supercategory Affine Brauer algebras (Regular) monomials. An example. Algebraically, it is written as y 2 1 y 4 4 b 4 s 1 s 3 s 6 y 1 y 2 6 y 7 s 5 b ∗ 2 b 2 b ∗ 3 . Our affine VW superalgebra sV V a is: ◮ super (signed) version of the degenerate BMW algebra, ◮ the signed version of the affine VW algebra, and ◮ an affine version of the Brauer superalgebra. Mee Seong Im West Point, NY 16

On the affine VW supercategory The center of affine VW superalgebras The center of sV V a . Theorem The center Z ( sV V a ) consists of all polynomials of the form � (( y i − y j ) 2 − 1) � f + c , 1 ≤ i < j ≤ a f ∈ C [ y 1 , . . . , y a ] S a and c ∈ C . where � The deformed squared Vandermonde determinant � 1 ≤ i < j ≤ a (( y i − y j ) 2 − 1) is symmetric, so � (( y i − y j ) 2 − 1) ∈ C [ y 1 , . . . , y a ] S a . 1 ≤ i < j ≤ a Mee Seong Im West Point, NY 17