Towers of algebras categorify the Heisenberg double

Transcription

1 Towers of algebras categorify the Heisenberg double Joint with: Oded Yacobi (Sydney) Alistair Savage University of Ottawa Slides available online: AlistairSavage.ca Preprint: arxiv: Alistair Savage (Ottawa) Heisenberg double categorification December 7, / 33

2 Outline Summary: Modules over towers of algebras categorify the Heisenberg double. Examples: Heisenberg algebra, Weyl algebra, quasi-heisenberg algebra. Application: The ring of quasisymmetric functions is free over the ring of symmetric functions (new proof). Overview 1 What is categorification? 2 Heisenberg double 3 Towers of algebras 4 Categorification of the Heisenberg double 5 Examples/applications 6 The quasi-heisenberg algebra and applications to quasisymmetric functions Alistair Savage (Ottawa) Heisenberg double categorification December 7, / 33

3 What is categorification? Suppose M is a module for a ring R. We would like to find an abelian category M such that K 0 (M) ϕ = M (as Z-modules), where K 0 (M) is the Grothendieck group of M. Then, for each r R (or, for those r in a fixed generating set), we want an exact endofunctor F r of M such that we have a commutative diagram: K 0 (M) [Fr ] K 0 (M) ϕ M r M ϕ Here [F r ] denotes the map induced by F r on K 0 (M). Alistair Savage (Ottawa) Heisenberg double categorification December 7, / 33

4 What is categorification? We would also like isomorphisms of functors lifting the relations of R. For example, suppose we have a relation in R: rs = 2sr + 3. Then we would like isomorphisms of functors F r F s = (Fs F r ) 2 Id 3. Fruits of categorification Classes of objects (simple, indecomposable projective) give distinguished bases with positivity and integrality properties. Uncovers hidden structure in the algebra and its representation. Provides tools for studying the category M. Applications to topology. Alistair Savage (Ottawa) Heisenberg double categorification December 7, / 33

5 Notation Fix a commutative ring k. All Hopf algebras, etc. are over this ring. For a Hopf algebra we have: multiplication comultiplication η unit ε counit We use Sweedler notation for coproducts: (a) = (a) a (1) a (2) Alistair Savage (Ottawa) Heisenberg double categorification December 7, / 33

6 Graded connected Hopf algebras Definition (Graded connected Hopf algebra) A bialgebra H is graded connected if H = n N H n, where each H n is f.g. and free as a k-module, (H k H l ) H k+l and (H k ) k j=0 H j H k j for all k, l N, η(k) H 0 and ε(h k ) = 0 for k > 0, H 0 = k1 H. These axioms imply that H is a Hopf algebra with invertible antipode and so we also call it a graded connected Hopf algebra. Examples 1 Polynomial ring k[x]. 2 Ring Sym of symmetric functions in variables x 1, x 2, Ring QSym of quasisymmetric functions. 4 Ring NSym of noncommutative symmetric functions. Alistair Savage (Ottawa) Heisenberg double categorification December 7, / 33

7 Dual pairs of Hopf algebras Definition (Dual pair) We say (H +, H ) is a dual pair of Hopf algebras if H ± are graded connected Hopf algebras, we have a perfect Hopf pairing, : H H + k. Via this pairing, we identify H ± with the graded dual of H. Any a H + defines L a End H + by left multiplication Any x H defines R x End H by right multiplication. Taking adjoints yields R x End H +. So we have injective k-algebra homomorphisms H + End H +, a L a, H End H +, x R x. Note: H End H + gives the left regular action of H on H +. Alistair Savage (Ottawa) Heisenberg double categorification December 7, / 33

8 The Heisenberg double Definition (Heisenberg double) The Heisenberg double of H + is the algebra h = h(h +, H ) given by: h = H + H as k-modules. We write a#x for a x, viewed as an element of h. Multiplication is given by: (a#x)(b#y) = (x) a R x (1) (b)#x (2)y = (x),(b) x (1), b (2) ab (1) #x (2) y. Remark: The Heisenberg double is a twist of the Drinfeld quantum double by a right 2-cocycle (Lu, 1994). Alistair Savage (Ottawa) Heisenberg double categorification December 7, / 33

9 Fock space Definition (Fock space) The algebra h has a natural representation on H + given by (a#x)(b) = a R x (b), a, b H +, x H. This is called the lowest weight Fock space representation of h and we denote it F. Stone von Neumann type theorem 1 The representation F is faithful. 2 If k is a field, then F is irreducible. 3 Any representation of h generated by a lowest weight vacuum vector is isomorphic to F. Thus, we may identify h as the subalgebra of End H + generated by L a, R x, a H +, x H. Alistair Savage (Ottawa) Heisenberg double categorification December 7, / 33

10 Examples of the Heisenberg double Example (Weyl algebra) Graded Hopf algebra: Q[x] Heisenberg double: Weyl algebra x, x = x + 1 Fock space: Polynomial representation on Q[x], x L x, d dx Example (Heisenberg algebra) Graded Hopf algebra: Ring Sym of symmetric functions Heisenberg double: Infinite-dimensional Heisenberg algebra Fock space: Bosonic Fock space Example (Quasi-Heisenberg algebra) Graded Hopf algebra: Ring QSym of quasisymmetric functions Heisenberg double: Quasi-Heisenberg algebra Fock space: Natural action on QSym Alistair Savage (Ottawa) Heisenberg double categorification December 7, / 33

11 The Heisenberg algebra as a Heisenberg double To get a presentation of the Heisenberg double, we choose generators for H + and H. Then a presentation is given by: the relations between the chosen generators of H +, the relations between the chosen generators of H, the commutation relations between the generators of H + and the generators of H. The Heisenberg algebra is the Heisenberg double of Sym: h = H + #H, H + = H = Sym. To give a presentation of h, we choose generators of H + and H (i.e. two sets of generators of Sym). Alistair Savage (Ottawa) Heisenberg double categorification December 7, / 33

12 The Heisenberg algebra as a Heisenberg double Power sums The power sums generate Sym over Q. Let p 1, p 2,... be the power sum functions in H + = Sym. Let p 1, p 2,... be the power sum functions in H = Sym. We get a presentation of h Z Q where the relations are: Integral form p m p n = p n p m, p mp n = p np m, p mp n = p n p m + mδ m,n. The elementary and complete symmetric functions generate Sym over Z. Let e 1, e 2,... be the elementary symmetric functions in H + = Sym. Let h 1, h 2,... be the complete symmetric functions in H = Sym. We get a presentation of h (over Z) where the relations are: e m e n = e n e m, h mh n = h nh m, h me n = e n h m + e n 1 h m 1. Alistair Savage (Ottawa) Heisenberg double categorification December 7, / 33

13 Goal Goal: To categorify Heisenberg doubles and their Fock space representations. So we want: Outline: categories whose Grothendieck groups are isomorphic to Fock space (as Z-modules), functors lifting the action of the Heisenberg double, isomorphisms of functors lifting the defining relations of the Heisenberg algebra. Categorify the (graded connected) Hopf algebra H + and its dual H. Define the functors that categorify the Fock space representation on H +. Alistair Savage (Ottawa) Heisenberg double categorification December 7, / 33

14 Towers of algebras Definition (Tower of algebras) A (strong) tower of algebras is a graded algebra A = n N A n over C such that: 1 Each A n is a f.d. algebra (with internal multiplication) and A 0 = C. 2 The multiplication A m A n A m+n is a homomorphism of algebras for all m, n. 3 Each A m+n is a two-sided projective (A m A n )-module. 4 Induction is (twisted) right adjoint to restriction. 5 A Mackey-like isomorphism relating induction and restriction holds. Examples Group algebras of symmetric groups: A n = CS n. Hecke algebras at generic q or at roots of unity. 0-Hecke algebras. Nilcoxeter algebras. Alistair Savage (Ottawa) Heisenberg double categorification December 7, / 33

15 Module categories and their Grothendieck groups For an f.d. algebra B, let B-mod = category of f.g. left B-modules, B-pmod = category of f.g. projective left B-modules. Let G 0 (B) = Grothendieck group of B-mod, K 0 (B) = Grothendieck group of B-pmod. There is a natural pairing, : K 0 (B) G 0 (B) Z, [P], [M] = dim k Hom B (P, M). If B is a subalgebra of A, we have: Induction: If N is a B-module, Ind A B N := A B N is an A-module. Restriction: If M is an A-module, Res A B M := Hom A(A, M) is a B-module. Alistair Savage (Ottawa) Heisenberg double categorification December 7, / 33

16 Grothendieck groups of towers of algebras Let A be a tower of algebras. Let G(A) = n N G 0 (A n ) and K(A) = n N K 0 (A n ). We have a pairing, : K(A) G(A) Z given by { dim C Hom An (P, M) if P A n -pmod, M A n -mod, [P], [M] = 0 otherwise. We have a Hopf algebra structure on G(A) (or K(A)) induced by (Am An)-mod : M N Ind Am+n A m A n (M N), An-mod : N Res An A k A l N, k+l=n η : Vect A 0 -mod, V V { V Vect if n = 0, ε An-mod : V 0 otherwise. Alistair Savage (Ottawa) Heisenberg double categorification December 7, / 33

17 Grothendieck groups of towers of algebras The functors,, η, ε are all exact. and K(A). So they induce operators on the Grothendieck groups G(A) Proposition (cf. N. Bergeron Li 2009) Under the above operations, (G(A), K(A)) is a dual pair of Hopf algebras. Distinguished bases The classes of the simple objects give a distinguished basis of G(A). The classes of the indecomposable projectives give a distinguished basis of K(A). These bases have positive integral structure coefficients. Alistair Savage (Ottawa) Heisenberg double categorification December 7, / 33

18 Examples Example (Nilcoxeter algebras) G(A) = Z[x], distinguished basis: x n, n N K(A) = Z[x, x 2 /2!, x 3 /3!,... ], distinguished basis: x n /n!, n N Example (Groups algebras of symmetric groups) G(A) = K(A) = Sym, distinguished basis: Schur functions The same holds for Hecke algebras at a generic parameter. Example (0-Hecke algebras) G(A) = QSym, distinguished basis: fundamental quasisymmetric functions K(A) = NSym, distinguished basis: noncommutative ribbon Schur functions Alistair Savage (Ottawa) Heisenberg double categorification December 7, / 33

19 The Heisenberg double associated to a tower of algebras Definition (Heisenberg double associated to a tower) To a tower of algebras A, we can now associate the Heisenberg double h(a) := h(g(a), K(A)) = G(A)#K(A) and its Fock space F(A) = G(A). Alistair Savage (Ottawa) Heisenberg double categorification December 7, / 33

20 Categorification of the Heisenberg double We wish to categorify the Fock space representation of h(a). We have already categorified the underlying Z-modules: G(A) = n N G 0 (A n ). We now need define functors on G(A) that lift the action of h(a) on Fock space. We would also like to have isomorphisms of functors that lift the defining relations of the Heisenberg double. Alistair Savage (Ottawa) Heisenberg double categorification December 7, / 33

21 Categorification of the Heisenberg double Consider the direct sum of categories: A-mod = n N A n -mod. For M A m -mod, we have the functor Ind M : A-mod A-mod, Ind M (N) = Ind Am+n A m A n (M N). For P A p -pmod, we have the functor Res P : A-mod A-mod, Res P (N) = Hom Ap (P, Res An A n p A p N). These functors are exact and so they induce endomorphisms of G(A). [Ind M ] and [Res P ] Alistair Savage (Ottawa) Heisenberg double categorification December 7, / 33

22 Categorification of the Heisenberg double Theorem (S. Yacobi 2013) The functors Ind M and Res P for M A-mod and P A-pmod categorify the Fock space representation F(A) of h(a). More precisely, for all M, N A-mod and P, Q A-pmod, we have isomorphisms of functors Ind M Ind N = Ind (M N), Res P Res Q = Res (P Q), Res P Ind M = Res (P) (M ). Thus, on the level of Grothendieck groups, we have ([M]#[P])([N]) = [Ind M ] [Res P ]([N]) = [Ind M Res P (N)] G(A). Note: For simplicity, we have assumed that induction is biadjoint to restriction above (i.e. no twisting is needed). Alistair Savage (Ottawa) Heisenberg double categorification December 7, / 33

23 Application: symmetric groups and the Heisenberg algebra Let S n be the symmetric group on n letters. Then A = n N ks n is a tower of algebras. We have K(A) = G(A) = Sym is the ring of symmetric functions, h(a) is the classical (infinite rank) Heisenberg algebra, F(A) is the usual (bosonic) Fock space representation, Stone von Neumann type theorem is the actual Stone von Neumann Theorem. Note: We get the same result if we replace ks n by the Hecke algebra H n (q) for generic q. So we get a categorification of the Heisenberg algebra. This recovers results of Geissinger and Zelevinsky. Alistair Savage (Ottawa) Heisenberg double categorification December 7, / 33

24 Application: symmetric groups and the Heisenberg algebra Distinguished basis of Fock space The simple modules and the indecomposable projective modules are both the Spect modules. These modules descend to the Schur functions in the Grothendieck group. Categorification of relations Recall that our integral form of the Heisenberg algebra had generators {e n, h n} n N+ and relations e m e n = e n e m, h mh n = h nh m, h me n = e n h m + e n 1 h m 1. If E n and L n are the sign and trivial reps of S n, respectively, then Ind Em Ind En = IndEn Ind Em, Res Lm Res Ln = ResLn Res Lm, Res Lm Ind En = (IndEn Res Lm ) ( Ind En 1 Res Lm 1 ). Alistair Savage (Ottawa) Heisenberg double categorification December 7, / 33

25 Application: symmetric groups and the Heisenberg algebra So we see that we obtain a categorification of the presentation in terms of the {e n, h n} n N+. By considering other appropriate modules, we could get a categorification of any presentation. For example: generators {h n, e n} n N+, generators {e n, e n} n N+, generators {h n, h n} n N+, etc. The main theorems are presentation independent. Alistair Savage (Ottawa) Heisenberg double categorification December 7, / 33

26 Application: nilcoxeter algebras and the Weyl algebra Let N n be the nilcoxeter algebra on n strands. Then N = n N N n is a tower of algebras. We have K(N) = Z[x], G(N) = Z[x, x 2 /2!, x 3 /3!,... ], the element x K(N) acts on G(N) via R x = x, h(n) is the subalgebra of End Z[x, x 2 /2!, x 3 /3!,... ] generated by x, x 2 /2!, x 3 /3!,... and x Thus we get a categorification of the Weyl algebra and its polynomial representation. This recovers a result of Khovanov. Alistair Savage (Ottawa) Heisenberg double categorification December 7, / 33

27 Application: nilcoxeter algebras and the Weyl algebra Distinguished basis of Fock space N n has a unique simple module L n of dimension one. Its projective cover is N n. So we get distinguished bases: G(N) = Z[x, x 2 /!, x 3 /3!,... ], [L n ] x n /n!, K(N) = Z[x], [N n ] x n. Categorifiation of relations If we let L be the trivial N 1 -module, then we have Res L Ind L = (IndL Res L ) Id. This is a categorification of the defining relation x = x + 1. Alistair Savage (Ottawa) Heisenberg double categorification December 7, / 33

28 Application: Hecke algebras at roots of unity Let A n be the Hecke algebra (of type A) specialized at an l-th root of unity. Then A = n N A n is a tower of algebras. Let J l Sym be the ideal generated by the power sums p l, p 2l,.... We have K(A) = Jl and G(A) = Sym/J l, h(a) is an integral form of the Heisenberg algebra: h(a) Z Q is the Heisenberg algebra (over Q), Questions: Work out minimal representations of these integral forms. Is h proj (A) = h(a)? Note: A-pmod categorifies the basic rep of ŝl n via i-induction and i-restriction. The above is a categorification of the action of the principal Heisenberg subalgebra of ŝl n. Alistair Savage (Ottawa) Heisenberg double categorification December 7, / 33

29 Quasisymmetric and noncommutative symmetric functions Let QSym be the algebra of quasisymmetric functions. Thus QSym is the subalgebra of Z x 1, x 2,... consisting of shift invariant functions. That is, f QSym if and only if coeff of x n 1 1 x n 2 2 x n k k in f = coeff of x n 2 i 1 x n 2 i 2 x n k i k in f for all 0 < i 1 < i 2 < < i k and n 1, n 2,..., n k N. Let NSym be the algebra of noncommutative symmetric functions. Thus NSym is the free associative algebra generated by h 1, h 2,.... Note: QSym and NSym are dual Hopf algebras. Alistair Savage (Ottawa) Heisenberg double categorification December 7, / 33

30 The quasi-heisenberg algebra Let H n (0) be the 0-Hecke algebra on n strands. Then A = n N H n(0) is a tower of algebras and K(A) = NSym and G(A) = QSym. Definition (Quasi-Heisenberg algebra) We call q := h(a) the quasi-heisenberg algebra. Alistair Savage (Ottawa) Heisenberg double categorification December 7, / 33

31 Application: QSym is free over Sym QSym is the Fock space for q. Using an inductive argument involving the Stone von Neumann type theorem, one can prove: Theorem (Hazewinkel 2001, S.-Yacobi 2013) The space QSym of quasisymmetric functions is free as a Sym-module. Remark The traditional proof of this result (due to Hazewinkel) involves constructing an explicit basis using modified Lyndon words. The new proof is representation theoretic and follows from the Stone von Neumann type theorem. Alistair Savage (Ottawa) Heisenberg double categorification December 7, / 33

32 Future directions Gradings Many examples of categorification in the literature involve gradings (e.g. categories of graded modules over a graded algebra). Often the passage from the non-graded to the graded setting results in a deformation of the algebraic object that is categorified. Goal: Generalize the above results to the graded setting (ongoing work with Daniele Rosso). Alistair Savage (Ottawa) Heisenberg double categorification December 7, / 33

33 Future directions Strong/graphical categorification The categorification of the Heisenberg algebra by Khovanov (q-deformed by Licata-S.) is stronger than the result obtained as a special case of the above main theorem (for the tower of symmetric groups). Khovanov defines a graphical monoidal category whose Grothendieck group recovers the Heisenberg algebra. Essentially, Khovanov s graphical category gives a precise description of the natural transformations involved in the isomorphisms of functors lifting the relations in the Heisenberg algebra. Goal: Generalize this graphical approach to more general towers (ongoing work with Oded Yacobi). Alistair Savage (Ottawa) Heisenberg double categorification December 7, / 33