A construction of the affine VW supercategory

Transcription

1 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, / 40

2 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, / 40

3 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, / 40

4 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 and super Jacobi identity for x, y, and z homogeneous. [x, y] = xy ( 1) xȳ yx = ( 1) xȳ [y, x] [x, [y, z]] = [[x, y], z] + ( 1) xȳ [y, [x, z]], 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, / 40

5 Preliminaries Matrix representation for gl(m n). {( ) A B gl(m n) := C D where M i,j := M i,j (C). : A M m,m, B, C t M m,n, D M n,n }, Since gl(m n) = gl(m n) 0 gl(m n) 1, gl(m n) 0 = {( A 0 0 D )} and gl(m n) 1 = {( 0 B 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, / 40

6 Periplectic Lie superalgebras p(n) Periplectic Lie superalgebras p(n). Let m = n. Then V = C 2n = 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, / 40

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

8 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 x, (counit) ɛ(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, / 40

9 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, v β(v, ), identifying V 1 with V 0 and V 0 with V 1. This induces the dual map (where β = β = 1) β : C = C (V V ) = V V, β (1) = i v i v i v i v i. Mee Seong Im The affine VW supercategory July 25, / 40

10 Periplectic Lie superalgebras p(n) Quadratic (fake) Casimir Ω & Jucys-Murphy elements y l s. Now, define Ω := 2 x X x x p(n) gl(n n) ( 2Ω = + ), 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, / 40

11 Periplectic Lie superalgebras p(n) We define l 1 Y l : M V a M V a as Y l := Ω i,l =, 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. i=0 Mee Seong Im The affine VW supercategory July 25, / 40

12 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, / 40

13 Schur-Weyl duality Classical Schur-Weyl duality. Consider the natural representations (CS a ) op φ End C (W a ψ ) and GL(W ) End C (W a ). Then Schur-Weyl duality gives us 1 φ(cs 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 CSa (W a ), 4 there is an irreducible (GL(W ), (CS a ) op )-bimodule decomposition (see next slide): Mee Seong Im The affine VW supercategory July 25, / 40

14 Schur-Weyl duality Classical Schur-Weyl duality (continued). where W a = λ=(λ 1,λ 2,...) a l(λ) n λ S λ, λ is an irreducible GL(W )-module associated to the partition λ, S λ is an irreducible CS 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 CS a and GL(W ) in End C (W a ) are centralizers of one another. Mee Seong Im The affine VW supercategory July 25, / 40

15 Schur-Weyl duality Other cases of Schur-Weyl duality. For the orthogonal group O(n) and symplectic group Sp 2n, the symmetric group S n should be replaced by a Brauer algebra. A Brauer algebra Br (x) a with a parameter x C is a unital C-algebra with generators s 1,..., s a 1, e 1,..., e a 1 and relations: s 2 i = 1, e 2 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, / 40

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

17 Schur-Weyl duality Now, in higher Schur-Weyl duality, we construct a result analogous to CS a = EndGL(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, / 40

18 Schur-Weyl duality The Hecke algebra H a contains the symmetric algebra CS a and the polynomial algebra C[y 1,..., y a ] as subalgebras. So as a vector space, H a = CSa C[y 1,..., y a ], and has a basis B = {wy k 1 1 yka a : w S a, k i N 0 }. Our goal: construct higher Schur-Weyl duality for p(n). That is, construct another algebra whose action on M V a commutes with the action of p(n). This algebra is precisely the degenerate affine Brauer superalgebra s V a. Mee Seong Im The affine VW supercategory July 25, / 40

19 Degenerate affine Brauer superalgebras Degenerate affine Brauer superalgebras (generators and local moves). s V a has generators s i, b i, b i, y j, where i = 1,..., a 1, j = 1,..., a and relations = = = = = Continued in the next slide. Mee Seong Im The affine VW supercategory July 25, / 40

20 Degenerate affine Brauer superalgebras Degenerate affine Brauer superalgs (local moves). = = = (braid reln) = (braid reln) = (adjunction) = (adjunction) = (untwisting reln) = = (untwisting reln) = Mee Seong Im The affine VW supercategory July 25, / 40

21 Degenerate affine Brauer superalgebras Degenerate affine Brauer superalgs (local moves). = = = = = = = + = = = Lemma. For any k 0, k = 0. Mee Seong Im The affine VW supercategory July 25, / 40

22 Normal diagrams. Degenerate affine Brauer superalgebras Call a diagram d Hom sbr (a, b) normal if all of the following hold: any two strings intersect at most once; no string intersects itself; no two cups or caps are at the same height; all cups are above all caps; the height of caps decreases when the caps are ordered from left to right with respect to their left ends; the height of cups increases when the cups are ordered from left to right with respect to their left ends. Every string in a normal diagram has either one cup, or one cap, or no cups and caps, and there are no closed loops. A diagram with no loops in Hom sbr (a, b) has a+b 2 strings. In particular, if a + b is odd then the Hom-space is zero. Mee Seong Im The affine VW supercategory July 25, / 40

23 Degenerate affine Brauer superalgebras Example: normal diagram in the signed Brauer algebra sbr a. Algebraically, it is written as s 2 s 3 s 5 b 2 b 2b 4 b 4s 1 s 3 s 6. The monomial corresponding to a normal diagram is called a regular monomial. Mee Seong Im The affine VW supercategory July 25, / 40

24 Connectors. Degenerate affine Brauer superalgebras Each normal diagram d Hom sbr (a, b), where a, b N 0, gives rise to a partition P (d) of the set of a + b points into 2-element subsets given by the endpoints of the strings in d. We call such a partition a connector, and write Conn(a, b) as the set of all such connectors. Its size is (a + b 1)!!. Mee Seong Im The affine VW supercategory July 25, / 40

25 Degenerate affine Brauer superalgebras Example. Let a = b = 2. Label the endpoints along the bottom row of d as 1 and 2 (reading from left to right), and label the endpoints along the top row of d as 1 and 2 (reading from left to right). Then 1 2 Hom sbr (2, 2) =, 1 2 }{{} d I } {{ } d s, }{{} d e Three possible connectors for a diagram in Hom sbr (2, 2): P (d I ) = {{1, 1}, {2, 2}}, P (d s ) = {{1, 2}, {2, 1}}, P (d e ) = {{1, 2}, { 1, 2}}, and Conn(2, 2) = {P (d I ), P (d s ), P (d e )}. Mee Seong Im The affine VW supercategory July 25, / 40

26 Degenerate affine Brauer superalgebras For each connector c Conn(a, b), we pick a normal diagram d c P 1 (c) Hom sbr (a, b). Remark. Different normal diagrams in a single fibre P 1 (c) differ only by braid relations, and thus represent the same morphism. Mee Seong Im The affine VW supercategory July 25, / 40

27 Theorem (BDEHHILNSS) The set S a,b = {d c : c Conn(a, b)} is a basis of Hom sbr (a, b). A dotted diagram d Hom s V (a, b) is normal if: the underlying diagram obtained by erasing the dots is normal; all dots on cups and caps are on the leftmost end, and all dots on the through strings are at the bottom. Example. A normal diagram in Hom s V (7, 7): Algebraically, it is written as y 4 2 s 2s 3 s 5 b 2 b 2b 4 b 4s 1 s 3 s 6 y 1 y 3 2 y 3y 6. Mee Seong Im The affine VW supercategory July 25, / 40

28 Normal dotted diagrams. Let S a,b be the normal dotted diagrams obtained by taking all diagrams in S a,b and adding dots to them in all possible ways. Let S k a,b S a,b be the diagrams with at most k dots. Theorem (Basis theorem, BDEHHILNSS) The set S k a,b is a basis of Hom s V (a, b) k, and the set S a,b is a basis of Hom s V (a, b). Our affine VW superalgebra s 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 The affine VW supercategory July 25, / 40

29 The center of affine VW superalgebras The center of s V a = End sv (a), a 2 N. Theorem (BDEHHILNSS) The center Z(s V a ) consists of all polynomials of the form ((y i y j ) 2 1) f + c, 1 i<j a where f C[y 1,..., y a ] Sa and c C. The deformed squared Vandermonde determinant 1 i<j a ((y i y j ) 2 1) is symmetric, so 1 i<j a ((y i y j ) 2 1) C[y 1,..., y a ] Sa. Mee Seong Im The affine VW supercategory July 25, / 40

30 Affine VW supercategory and Brauer supercategory Affine VW supercategory s V and connections to Brauer supercategory sbr. The affine VW supercategory (or the affine Nazarov-Wenzl supercategory) is the C-linear strict monoidal supercategory generated as a monoidal supercategory by a single object, morphisms s = :, = : 1, = : 1, and an additional morphism y = :, subject to the braid, snake (adjunction), and untwisting relations, and the dot relations: = + = +. Objects in s V can be identified with natural numbers, identifying a N 0 with a, 0 = 1, and the morphisms are linear combinations of dotted diagrams. Mee Seong Im The affine VW supercategory July 25, / 40

31 Affine VW supercategory and Brauer supercategory s V and sbr. The category s V can alternatively be generated by vertically stacking i, i, s i, and y i = 1 i 1 y 1 a i Hom s V (a, a). It is a filtered category, i.e., the hom spaces Hom s V (a, b) have a filtration by the span Hom s V (a, b) k of all dotted diagrams with at most k dots. The Brauer supercategory sbr is the C-linear strict monoidal supercategory generated as a monoidal supercategory by a single object, and morphisms s = :, = : 1, and = : 1, subject to the relations above. If M is the trivial representation, then actions on s V factor through sbr. Mee Seong Im The affine VW supercategory July 25, / 40

32 Affine VW supercategory and Brauer supercategory Thank you. Questions? Mee Seong Im The affine VW supercategory July 25, / 40

33 Affine VW supercategory and Brauer supercategory The algebra A h and its specializations A t, where t C. Definition Let A h be the superalgebra over C[ h] with generators s i, e i, y j for 1 i a 1, 1 j a, where s i = e i = y j = 0, subject to the relations: 1 Involutions: s 2 i = 1 for 1 i < a. 2 Commutation relations: 1 s i e j = e j s i if i j > 1, 2 e i e j = e j e i if i j > 1, 3 e i y j = y j e i if j i, i + 1, 4 y i y j = y j y i for 1 i, j a. 3 Affine braid relations: 1 s i s j = s j s i if i j > 1, 2 s i s i+1 s i = s i+1 s i s i+1 for 1 i a 1, 3 s i y j = y j s i if j i, i Snake relations: 1 e i+1 e i e i+1 = e i+1, 2 e i e i+1 e i = e i for 1 i a 2. 5 Tangle and untwisting relations: 1 e i s i = e i and s i e i = e i for 1 i a 1, 2 s i e i+1 e i = s i+1 e i, 3 s i+1 e i e i+1 = s i e i+1, 4 e i+1 e i s i+1 = e i+1 s i, 5 e i e i+1 s i = e i s i+1 for 1 i a 2. 6 Idempotent relations: e 2 i = 0 for 1 i a 1. 7 Skein relations: 1 s i y i y i+1 s i = he i h, 2 y i s i s i y i+1 = he i h for 1 i a 1. 8 Unwrapping relations: e 1 y k 1 e 1 = 0 for k N. 9 (Anti)-symmetry relations: 1 e i (y i+1 y i ) = he i, 2 (y i+1 y i )e i = he i for 1 i a 1. For t C, let A t be the quotient of A h by the ideal generated by h t. Mee Seong Im The affine VW supercategory July 25, / 40

34 A sketch of proof of the Theorem on slide 27 A sketch of proof of the Theorem on slide The filtered algebra s V a (via the filtration by the degree of the polynomials in C[y 1,..., y a ]) is a Poincaré-Birkhoff-Witt (PBW) deformation of the associated graded superalgebra gs V a = gr(s V a ), 2 For h a parameter, the Rees construction gives the algebra A h over C[ h] such that the specializations h = 1 and h = 0 are precisely A 1 = s V a and A 0 = gs V a, 3 Describe the center of the C[ h]-algebra A h, and all its specializations A t for any t C using the Basis Theorem, 4 Determine the center of gs V a using the isomorphism Rees(Z(A 1 )) = Z(Rees(A 1 )) = Z(A h ), and 5 Find a lift of the appropriate basis elements to s V a to obtain the center of s V a. Mee Seong Im The affine VW supercategory July 25, / 40

35 A sketch of proof of the Theorem on slide 27 Expanding on 2. Let B = k 0 B k be a filtered C-algebra. The Rees algebra of B is the C[ h]-algebra Rees(B), given as a C-vector space by Rees(B) = k 0 B k h k, with multiplication and the h-action given by (a h i )(b h j ) = (ab) h i+j for a B i, b B j, and ab B i+j, the product in B. It is graded as a C-algebra by the powers of h. Lemma 1 Let i 0 S i be a basis of B compatible with the filtration, where S i s are pairwise disjoint, and k i=0 S i is a basis of B k. Then i 0 S i h i is a C[ h]-basis of Rees(B). 2 Z(Rees(B)) = Rees(Z(B)). 3 Rees(A 1 ) = A h, an isomorphism of C[ h]-algebras. Mee Seong Im The affine VW supercategory July 25, / 40

36 A sketch of proof of the Theorem on slide 27 Expanding on 3. Show that Z(A h ) C[ h][y 1,..., y a ] Sa. Lemma For f A h, the following are equivalent: (a) fy i = y i f for all i [a] = {1, 2,..., a}; (b) f C[ h][y 1,..., y a ]. So Z(A h ) C[ h][y 1,..., y a ]. Lemma. Let f C[ h][y 1,..., y a ] A h and 1 i a 1. (a) (b) If fs i = s i f, then f(y 1,..., y i, y i+1,..., y a ) = f(y 1,..., y i+1, y i,..., y a ). For the special value h = 0, the converse also holds: if f(y 1,..., y i, y i+1,..., y a ) = f(y 1,..., y i+1, y i,..., y a ), then fs i = s i f in A 0. So Z(A h ) is a subalgebra of C[ h][y 1,..., y a ] Sa. Mee Seong Im The affine VW supercategory July 25, / 40

37 A sketch of proof of the Theorem on slide 27 Expanding on 3 (continued). Consider the following elements in C[ h][y 1,..., y a ]: z ij = (y i y j ) 2, for 1 i j a and D h = where D h is symmetric. So D h C[ h][y 1,..., y a ] Sa. Use D h to produce central elements in A h. Lemma 1 i<j a (z ij h 2 ), 1 For any 1 i a 1, e i (z i,i+1 h 2 ) = (z i,i+1 h 2 ) e i = 0 in A h, and consequently e i D h = D h e i = 0. 2 For any 1 k a 1, we have D h s k = s k D h. 3 Let 1 i a 1, and let f C[ h][y 1,..., y a ] be symmetric in y i, y i+1. Then there exist polynomials p j = p j (y 1,..., y a ) C[ h][y 1,..., y a ] such that fs i = s i f + deg f 1 j=0 y j i e i p j. Mee Seong Im The affine VW supercategory July 25, / 40

38 A sketch of proof of the Theorem on slide 27 Expanding on 3 (continued). Lemma Let f C[ h][y 1,..., y a ] Sa be an arbitrary symmetric polynomial, and c C. Then f = D h f + c Z(A h ). Expanding on 4. Proposition. The center Z(A 0 ) of the graded VW superalgebra gs V a consists of all f C[y 1,..., y a ] of the form f = D 0 f + c, for f C[y1,..., y a ] S a and c C. Mee Seong Im The affine VW supercategory July 25, / 40

39 A sketch of proof of the Theorem on slide 27 Expanding on 5. Theorem (BDEHHILNSS) The center Z(s V a ) of the VW superalgebra s V a = A 1 consists of all f C[y 1,..., y n ] of the form f = D 1 f + c, for an arbitrary symmetric polynomial f C[y1,..., y a ] Sa and c C. Proof. For any filtered algebra B there exists a canonical injective algebra homomorphism ϕ : gr Z(B) Z(gr(B)), given by ϕ(f + Z(B) (k 1) ) = f + B (k 1) for f Z(B) k. For B = s V a and gr(b) = gs V a, Z(A 0 ) consists of elements of the form f = D 0 f + c for f a symmetric polynomial and c a constant. Since D 1 f + c Z(s Va ), we have ϕ(c) = c, and for f symmetric and homogeneous of degree k, ϕ(d 1 f + s V a a(a 1)+k 1 ) = D 0 f. Using the above Proposition, we see that every f Z(gs V a ) is in the image of ϕ, so ϕ is an isomorphism. Mee Seong Im The affine VW supercategory July 25, / 40

40 A sketch of proof of the Theorem on slide 27 Expanding on 5 (continued). Theorem (BDEHHILNSS) The center Z(A h ) of the superalgebra A h consists of polynomials f C[ h][y 1,..., y n ] of the form f = D h f + c, for an arbitrary symmetric polynomial f C[ h][y1,..., y a ] Sa and c C[ h]. Proof. The center Z(A h ) is isomorphic to Z(Rees(A 1 )), which is also isomorphic to Rees(Z(A 1 )). The center Z(A 1 ) consists of elements of the form f = D 1 f + c, with f C[y1,..., y a ] Sa and c C. Assume f is k+a(a 1) homogeneous of degree k. Then D 1 f A 1, which gives an element D 1 f h k+a(a 1) of Rees(Z(A 1 )) = Z(Rees(A 1 )). We see that Z(A h ) is spanned by constants and the preimages under the isomorphism A h = Rees(A1 ) of elements D 1 f h k+a(a 1), which are equal to D h f. Mee Seong Im The affine VW supercategory July 25, / 40