A construction of the affine VW supercategory Mee Seong Im United - PowerPoint PPT Presentation

A construction of the affine VW supercategory Mee Seong Im United States Military Academy West Point, NY Institute for Computational and Experimental Research in Mathematics Brown University, Providence, RI Mee Seong Im The affine VW supercategory July 25, 2018 1 / 40

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 The affine VW supercategory July 25, 2018 2 / 40

Preliminaries Background: vector superspaces. Work over C . A Z 2 -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 denoted by 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 . Mee Seong Im The affine VW supercategory July 25, 2018 3 / 40

Preliminaries A Lie superalgebra is a Z 2 -graded vector space g = g ¯ 0 ⊕ g ¯ 1 with a Lie superbracket (supercommutator) [ , ] : g × g → g that satisfies super skew symmetry [ x, y ] = xy − ( − 1) ¯ x ¯ y yx = − ( − 1) ¯ x ¯ y [ y, x ] and super Jacobi identity [ x, [ y, z ]] = [[ x, y ] , z ] + ( − 1) ¯ x ¯ y [ y, [ x, z ]] , for x , y , and z homogeneous. Now, given a homogeneous ordered basis for V = C { v 1 , . . . , v m } ⊕ C { v 1 ′ , . . . , v n ′ } , � �� � � �� � V 0 V 1 the Lie superalgebra is the endomorphism algebra End C ( V ) explicitly given by Mee Seong Im The affine VW supercategory July 25, 2018 4 / 40

Preliminaries Matrix representation for gl ( m | n ) . �� 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 gl ( m | n ) is the general linear Lie superalgebra , and V is the natural representation of gl ( m | n ) . The grading on gl ( m | n ) is induced by V . Mee Seong Im The affine VW supercategory July 25, 2018 5 / 40

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 = C ¯ 0 as an odd, symmetric, nondegenerate bilinear form satisfying: β ( v, w ) = β ( w, v ) , β ( v, w ) = 0 if v = w. That is, β satisfies β ( v, w ) = ( − 1) ¯ v ¯ w β ( w, v ) . We define periplectic (strange) Lie superalgebras as: p ( n ) := { x ∈ End C ( V ) : β ( xv, w ) + ( − 1) xv β ( v, xw ) = 0 } . Mee Seong Im The affine VW supercategory July 25, 2018 6 / 40

Periplectic Lie superalgebras p ( n ) In terms of the above basis, �� A � � B ∈ gl ( n | n ) : B = B t , C = − C t p ( n ) = , − A t C where �� A �� �� 0 �� 0 B ∼ p ( n ) ¯ 0 = = gl n ( C ) and p ( n ) ¯ 1 = . − A t 0 C 0 Mee Seong Im The affine VW supercategory July 25, 2018 7 / 40

Periplectic Lie superalgebras p ( n ) Symmetric monoidal structure. Consider the category C of representations of p ( n ) , where 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 the universal enveloping algebra U ( p ( n )) is a Hopf superalgebra: (coproduct) ∆( x ) = x ⊗ 1 + 1 ⊗ x , (counit) ǫ ( x ) = 0 , (antipode) S(x) = -x. So C is a monoidal category. Now for x ⊗ y ∈ U ( p ( n )) ⊗ U ( p ( n )) on v ⊗ w , ( x ⊗ y ) . ( v ⊗ w ) = ( − 1) yv xv ⊗ yw. Mee Seong Im The affine VW supercategory July 25, 2018 8 / 40

Periplectic Lie superalgebras p ( n ) Symmetric monoidal structure. For x, y, a, b ∈ U ( p ( n )) , multiplication is defined as ( 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 whose dual satisfies σ ∗ = − σ . Thus C is a symmetric monoidal category. Furthermore, β induces an identification between V and its dual V ∗ via V → V ∗ , identifying V 1 with V ∗ 0 and V 0 with V ∗ v �→ β ( v, − ) , 1 . This induces the dual map (where β = β ∗ = 1 ) � β ∗ : C ∼ = C ∗ − → ( V ⊗ V ) ∗ ∼ β ∗ (1) = v i ′ ⊗ v i − v i ⊗ v i ′ . = V ⊗ V, i Mee Seong Im The affine VW supercategory July 25, 2018 9 / 40

Periplectic Lie superalgebras p ( n ) Quadratic (fake) Casimir Ω & Jucys-Murphy elements y ℓ ’s. Now, define � � � x ⊗ x ∗ ∈ p ( n ) ⊗ gl ( n | n ) Ω := 2 2Ω = + , x ∈X where X is a basis of p ( n ) and x ∗ ∈ p ( n ) ∗ is a dual basis element of p ( n ) , with p ( n ) ∗ = p ( n ) ⊥ , taken with respect to the supertrace: � A � B str = tr( A ) − tr( D ) . C D The actions of Ω and p ( n ) commute on M ⊗ V , so Ω is in the centralizer End p ( n ) ( M ⊗ V ) . Mee Seong Im The affine VW supercategory July 25, 2018 10 / 40

Periplectic Lie superalgebras p ( n ) 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 The affine VW supercategory July 25, 2018 11 / 40

Schur-Weyl duality Review: 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 the full linear group 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 The affine VW supercategory July 25, 2018 12 / 40

Schur-Weyl duality Classical Schur-Weyl duality. Consider the natural representations φ ψ ( C S a ) op → End C ( W ⊗ a ) and GL ( W ) → End C ( W ⊗ a ) . − − Then Schur-Weyl duality gives us φ ( C S a ) = End GL ( W ) ( W ⊗ a ) , 1 if n ≥ a , then φ is injective. So im φ ∼ = End GL ( W ) ( W ⊗ a ) , 2 ψ ( GL ( W )) = End C S a ( W ⊗ a ) , 3 there is an irreducible ( GL ( W ) , ( C S a ) op ) -bimodule decomposition 4 (see next slide): Mee Seong Im The affine VW supercategory July 25, 2018 13 / 40

Schur-Weyl duality Classical 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 the partition λ , S λ is an irreducible C S a (Specht) module associated to λ , and ℓ ( λ ) = max { i ∈ Z : λ i � = 0 , λ = ( λ 1 , λ 2 , . . . ) } . In the above setting, we say C S a and GL ( W ) in End C ( W ⊗ a ) are centralizers of one another. Mee Seong Im The affine VW supercategory July 25, 2018 14 / 40

Schur-Weyl duality Other cases of Schur-Weyl duality. For the orthogonal group O ( n ) and symplectic group Sp 2 n , the symmetric group S n should be replaced by a Brauer algebra. A Brauer algebra Br ( x ) with a parameter x ∈ C is a unital C -algebra a with generators s 1 , . . . , s a − 1 , e 1 , . . . , e a − 1 and relations: s 2 e 2 i = 1 , i = xe i , e i s i = e i = s i e i for all 1 ≤ i ≤ a − 1 , s i s j = s j s i , s i e j = e j s i , e i e j = e j e i for all 1 ≤ i < j − 1 ≤ a − 2 , s i s i +1 s i = s i +1 s i s i +1 for all 1 ≤ i ≤ a − 2 , e i e i +1 e i = e i , e i +1 e i e i +1 = e i +1 for all 1 ≤ i ≤ a − 2 , s i e i +1 e i = s i +1 e i , e i +1 e i s i +1 = e i +1 s i for all 1 ≤ i ≤ a − 2 . Mee Seong Im The affine VW supercategory July 25, 2018 15 / 40

Schur-Weyl duality The group ring of the Brauer algebra Br ( n ) and O ( n ) in End( W ⊗ a ) a centralize one another, where dim W = n , and the group ring of the Brauer algebra Br ( − 2 n ) and Sp 2 n in End( V ⊗ a ) a centralize one another, where dim V = 2 n . Mee Seong Im The affine VW supercategory July 25, 2018 16 / 40

Schur-Weyl duality Now, 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 , where H a is the degenerate affine Hecke algebra , i.e., it is a deformation of the symmetric group S a . The algebra H a has generators s 1 , . . . , s a − 1 , y 1 , . . . , y a and relations s 2 i = 1 , s i s j = s j s i whenever | i − j | > 1 , s i s i +1 s i = s i +1 s i s i +1 , y i y j = y j y i , y i s j = s j y i whenever i − j � = 0 , 1 , y i +1 s i = s i y i + 1 . Mee Seong Im The affine VW supercategory July 25, 2018 17 / 40