Categorification and Heisenberg doubles arising from towers of algebras
aa r X i v : . [ m a t h . R T ] S e p CATEGORIFICATION AND HEISENBERG DOUBLES ARISING FROMTOWERS OF ALGEBRAS
ALISTAIR SAVAGE AND ODED YACOBI
Abstract.
The Grothendieck groups of the categories of finitely generated modules andfinitely generated projective modules over a tower of algebras can be endowed with (co)algebrastructures that, in many cases of interest, give rise to a dual pair of Hopf algebras. More-over, given a dual pair of Hopf algebras, one can construct an algebra called the Heisenbergdouble, which is a generalization of the classical Heisenberg algebra. The aim of this paperis to study Heisenberg doubles arising from towers of algebras in this manner. First, we de-velop the basic representation theory of such Heisenberg doubles and show that if inductionand restriction satisfy Mackey-like isomorphisms then the Fock space representation of theHeisenberg double has a natural categorification. This unifies the existing categorificationsof the polynomial representation of the Weyl algebra and the Fock space representationof the Heisenberg algebra. Second, we develop in detail the theory applied to the towerof 0-Hecke algebras, obtaining new Heisenberg-like algebras that we call quasi-Heisenbergalgebras . As an application of a generalized Stone–von Neumann Theorem, we give a newproof of the fact that the ring of quasisymmetric functions is free over the ring of symmetricfunctions.
Contents
1. Introduction 22. The Heisenberg double 43. Towers of algebras and categorification of Fock space 84. Hecke-like towers of algebras 165. Nilcoxeter algebras 206. Hecke algebras at generic parameters 217. Hecke algebras at roots of unity 228. 0-Hecke algebras 249. Application: QSym is free over Sym 27References 29
Mathematics Subject Classification.
Key words and phrases.
Heisenberg double, tower of algebras, categorification, Hopf algebra, Hecke alge-bra, quasisymmetric function, noncommutative symmetric function, Heisenberg algebra, Fock space.The first author was supported by a Discovery Grant from the Natural Sciences and Engineering ResearchCouncil of Canada. The second author was supported by an AMS–Simons Travel Grant. Part of this workwas completed while the second author was at the University of Toronto.
ALISTAIR SAVAGE AND ODED YACOBI Introduction
The interplay between symmetric groups and the Heisenberg algebra has a rich history,with implications in combinatorics, representation theory, and mathematical physics. A foun-dational result in this theory is due to Geissinger, who gave a representation theoretic real-ization of the bialgebra of symmetric functions Sym by considering the Grothendieck groupsof representations of all symmetric groups over a field k of characteristic zero (see [Gei77]).In particular, he constructed an isomorphism of bialgebrasSym ∼ = L ∞ n =0 K ( k [ S n ] -mod) , where K ( C ) denotes the Grothendieck group of an abelian category C . Multiplication isdescribed by the induction functor[Ind] : K ( k [ S n ] -mod) ⊗ K ( k [ S m ] -mod) → K ( k [ S n + m ] -mod) , while comultiplication is given by restriction. Mackey theory for induction and restrictionin symmetric groups implies that the coproduct is an algebra homomorphism. For each S n -module V , multiplication by the class [ V ] ∈ K ( k [ S n ] -mod) defines an endomorphism of L ∞ n =0 K ( k [ S n ] -mod). These endomorphisms, together with their adjoints, define a repre-sentation of the Heisenberg algebra on L ∞ n =0 K ( k [ S n ] -mod).Geissinger’s construction was q -deformed by Zelevinsky in [Zel81], who replaced the groupalgebra of the symmetric group k [ S n ] by the Hecke algebra H n ( q ) at generic q . Again,endomorphisms of the Grothendieck group given by multiplication by classes [ V ], togetherwith their adjoints, generate a representation of the Heisenberg algebra.The above results can be enhanced to a categorification of the Heisenberg algebra andits Fock space representation via categories of modules over symmetric groups and Heckealgebras. A strengthened version of this categorification, which includes information aboutthe natural transformations involved, was given in [Kho] for the case of symmetric groupsand in [LS13] for the case of Hecke algebras.The group algebras of symmetric groups and Hecke algebras are both examples of towersof algebras . A tower of algebras is a graded algebra A = L n ≥ A n , where each A n is itselfan algebra (with a different multiplication) and such that the multiplication in A induceshomomorphisms A m ⊗ A n → A m + n of algebras (see Definition 3.1). In addition to thosementioned above, examples include nilcoxeter algebras, 0-Hecke algebras, Hecke algebras atroots of unity, wreath products (semidirect products of symmetric groups and finite groups,see [FJW00, CL12]), group algebras of finite general linear groups, and cyclotomic Khovanov–Lauda–Rouquier algebras (quiver Hecke algebras). To a tower of algebras, one can associatethe Z -modules G ( A ) = L n K ( A n -mod) and K ( A ) = L n K ( A n -pmod), where A n -mod(respectively A n -pmod) is the category of finitely generated (respectively finitely generatedprojective) A n -modules. In many cases, induction and restriction endow K ( A ) and G ( A ) withthe structure of dual Hopf algebras. For example, in [BL09] Bergeron and Li introduced a setof axioms for a tower of algebras that ensure this duality (although the axioms we considerin the current paper are different).One of the main goals of the current paper is to generalize the above categorifications ofthe Fock space representation of the Heisenberg algebra to more general towers of algebras.We see that, in the general situation, the Heisenberg algebra h is replaced by the Heisenbergdouble (see Definition 2.6) of G ( A ). The Heisenberg double of a Hopf algebra is different ATEGORIFICATION AND HEISENBERG DOUBLES ARISING FROM TOWERS OF ALGEBRAS 3 from, but closely related to, the more well known Drinfeld quantum double. As a k -module,the Heisenberg double h ( H + , H − ) of a Hopf algebra H + (over k ) with dual H − is isomorphicto H + ⊗ k H − , and the factors H − and H + are subalgebras. The most well known exampleof this construction is when H − and H + are both the Hopf algebra of symmetric functions,which is self-dual. In this case the Heisenberg double is the classical Heisenberg algebra. Ingeneral, there is a natural action of h ( H + , H − ) on its Fock space H + generalizing the usualFock space representation of the Heisenberg algebra. Our first result (Theorem 2.11) is ageneralization of the well known Stone–von Neumann Theorem to this Heisenberg doublesetting.In the special case of dual Hopf algebras arising from a tower of algebras A , we denote theHeisenberg double by h ( A ) and the resulting Fock space by F ( A ). In this situation, there isa natural subalgebra of h ( A ). In particular, the image G proj ( A ) of the natural Cartan map K ( A ) → G ( A ) is a Hopf subalgebra of G ( A ), and we consider also the projective Heisenbergdouble h proj ( A ) which is, by definition, the subalgebra of h ( A ) generated by G proj ( A ) and K ( A ). Then h proj ( A ) acts on its Fock space G proj ( A ), and a Stone–von Neumann type theoremalso holds for this action (see Proposition 3.15).We then focus our attention on towers of algebras that satisfy natural compatibility condi-tions between induction and restriction analogous to the well known Mackey theory for finitegroups. We call these towers of algebras strong (see Definition 3.4) and give a necessary andsufficient condition for them to give rise to dual pairs of Hopf algebras (i.e. be dualizing ).Our central theorem (Theorem 3.18) is that, for such towers, the Fock spaces representa-tions of the algebras h ( A ) and h proj ( A ) admit categorifications coming from induction andrestrictions functors on L n A n -mod and L n A n -pmod respectively.To illustrate our main result, we apply it to several towers of algebras that are quotients ofgroup algebras of braid groups by quadratic relations (see Definition 4.1). We first show thatall towers of this form are strong and dualizing. Examples include the towers of nilcoxeteralgebras, Hecke algebras, and 0-Hecke algebras. Starting with the tower of nilcoxeter alge-bras, we recover Khovanov’s categorification of the polynomial representation of the Weylalgebra (see Section 5). Taking instead the tower of Hecke algebras at a generic parameter,we recover (weakened versions of) the categorifications of the Fock space representation ofthe Heisenberg algebra described by Khovanov and Licata–Savage (see Section 6). Consid-ering the tower of Hecke algebras at a root of unity, we obtain a different categorification ofthe Fock space representation of the Heisenberg algebra (Proposition 7.2) which, in the set-ting of the existing categorification of the basic representation of affine sl n using this tower,corresponds to the principal Heisenberg subalgebra. In this way, we see that Theorem 3.18provides a uniform treatment and generalization of these categorification results. A majorfeature of our categorification is that it does not depend on any presentation of the algebrasin question, in contrast to many categorification results in the literature.We explore the example of the tower A of 0-Hecke algebras in some detail. In this case, itis known that K ( A ) and G ( A ) are the Hopf algebras of noncommutative symmetric functionsand quasisymmetric functions respectively. However, the algebras h ( A ) and h proj ( A ), whichwe call the quasi-Heisenberg algebra and projective quasi-Heisenberg algebra , do not seem tohave been studied in the literature. We give presentations of these algebras by generatorsand relations (see Section 8.4). The algebra h proj ( A ) turns out to be particularly simple as it ALISTAIR SAVAGE AND ODED YACOBI is a “de-abelianization” of the usual Heisenberg algebra (see Proposition 8.7). As an appli-cation of the generalized Stone–von Neumann Theorem in this case, we give a representationtheoretic proof of the fact that the ring of quasisymmetric functions is free over the ringof symmetric functions (Proposition 9.2). Our proof is quite different than previous proofsappearing in the literature.There are many more examples of towers of algebras for which we do not work out thedetailed implications of our main theorem. Furthermore, we expect that the results of thispaper could be generalized to apply to towers of superalgebras. Examples of such towersinclude Sergeev algebras and 0-Hecke-Clifford algebras. We leave such generalizations forfuture work.
Notation.
We let N and N + denote the set of nonnegative and positive integers respectively.We let k be a commutative ring (with unit) and F be a field. For n ∈ N , we let P ( n ) denotethe set of all partitions of n , with the convention that P (0) = { ∅ } , and let P = S n ∈ N P ( n ).Similarly, we let C ( n ) denote the set of all compositions of n and let C = S n ∈ N C ( n ). For acomposition or partition α , we let ℓ ( α ) denote the length of α (i.e. the number of nonzeroparts) and let | α | denote its size (i.e. the sum of its parts). By a slight abuse of terminology,we will use the terms module and representation interchangeably. Acknowledgements.
The authors would like to thank N. Bergeron, J. Bernstein, M. Kho-vanov, A. Lauda, A. Licata, C. Reutenauer, J.-Y. Thibon, and M. Zabrocki for useful con-versations. 2.
The Heisenberg double
In this section, we review the definition of the Heisenberg double of a Hopf algebra andstate some important facts about its natural Fock space representation. In particular, weprove a generalization of the well known Stone–von Neumann Theorem (Theorem 2.11).We fix a commutative ring k and all algebras, coalgebras, bialgebras and Hopf algebraswill be over k . We will denote the multiplication, comultiplication, unit, counit and antipodeof a Hopf algebra by ∇ , ∆, η , ε and S respectively. We will use Sweedler notation∆( a ) = P ( a ) a (1) ⊗ a (2) for coproducts. For a k -module V , we will simply write End V for End k V . All tensorproducts are over k unless otherwise indicated.2.1. Dual Hopf algebras.
We begin by recalling the notion of dual (graded connected)Hopf algebras.
Definition 2.1 (Graded connected Hopf algebra) . We say that a bialgebra H is graded if H = L n ∈ N H n , where each H n is finitely generated and free as a k -module, and the followingconditions are satisfied: ∇ ( H k ⊗ H ℓ ) ⊆ H k + ℓ , ∆( H k ) ⊆ L kj =0 H j ⊗ H k − j , k, ℓ ∈ N ,η ( k ) ⊆ H , ε ( H k ) = 0 for k ∈ N + . We say that H is graded connected if it is graded and H = k H . Recall that a gradedconnected bialgebra is a Hopf algebra with invertible antipode (see, for example, [Haz08,p. 389, Cor. 5]) and thus we will also call such an object a graded connected Hopf algebra . ATEGORIFICATION AND HEISENBERG DOUBLES ARISING FROM TOWERS OF ALGEBRAS 5 If H = L n ∈ N H n is a graded bialgebra, then its graded dual L n ∈ N H ∗ n is also a gradedbialgebra. Remark 2.2.
In general, one need not assume that the H n are free as k -modules. Instead,one needs only assume that(2.1) H ∗ k ⊗ H ∗ ℓ ∼ = ( H k ⊗ H ℓ ) ∗ for all k, ℓ ∈ N in order for the graded dual to be a graded bialgebra. However, since our interest lies mainlyin dual Hopf algebras arising from towers of algebras, for which the H n are free as k -modules,we will make this assumption from the start (in which case (2.1) is automatically satisfied). Definition 2.3 (Hopf pairing) . If H and H ′ are Hopf algebras over k , then a Hopf pairing is a bilinear map h· , ·i : H × H ′ → k such that h ab, x i = h a ⊗ b, ∆( x ) i = P ( x ) h a, x (1) ih b, x (2) i , h a, xy i = h ∆( a ) , x ⊗ y i = P ( a ) h a (1) , x ih a (2) , y i , h H , x i = ε ( x ) , h a, H ′ i = ε ( a ) , for all a, b ∈ H , x, y ∈ H ′ . Note that such a Hopf pairing automatically satisfies h a, S ( x ) i = h S ( a ) , x i for all a ∈ H and x ∈ H ′ .Recall that, for k -modules V and W , a bilinear form h· , ·i : V × W → k is called a perfectpairing if the induced map Φ : V → W ∗ given by Φ( v )( w ) = h v, w i is an isomorphism. Definition 2.4 (Dual pair) . We say that ( H + , H − ) is a dual pair of Hopf algebras if H + and H − are both graded connected Hopf algebras and H ± is graded dual to H ∓ (as a Hopfalgebra) via a perfect Hopf pairing h· , ·i : H − × H + → k .2.2. The Heisenberg double.
For the remainder of this section, we fix a dual pair ( H + , H − )of Hopf algebras. Then any a ∈ H + defines an element L a ∈ End H + by left multiplication.Similarly, any x ∈ H − defines an element R x ∈ End H − by right multiplication, whose ad-joint R x ∗ is an element of End H + . (In the case that H + or H − is commutative, we oftenomit the superscript L or R .) In this way we have k -algebra homomorphisms H + ֒ → End H + , a L a, (2.2) H − ֒ → End H + , x R x ∗ . (2.3)The action of H − on H + given by (2.3) is called the left-regular action . Lemma 2.5.
The left-regular action of H − on H + is given by R x ∗ ( a ) = P ( a ) h x, a (2) i a (1) for all x ∈ H − , a ∈ H + . Proof.
For all x, y ∈ H − and a ∈ H + , we have h y, R x ∗ ( a ) i = h yx, a i = h y ⊗ x, ∆( a ) i = P ( a ) h y ⊗ x, a (1) ⊗ a (2) i = P ( a ) h y, a (1) ih x, a (2) i = h y, P ( a ) h x, a (2) i a (1) i . The result then follows from the nondegeneracy of the bilinear form. (cid:3)
ALISTAIR SAVAGE AND ODED YACOBI
It is clear that the map (2.2) is injective. The map (2.3) is also injective. Indeed, for x ∈ H − , choose a ∈ H + such that h x, a i 6 = 0. Then h , R x ∗ ( a ) i = h x, a i 6 = 0, and so R x ∗ = 0.Since H + = L n ∈ N H + n is N -graded, we have a natural algebra Z -grading End H + = L n ∈ Z End n H + . It is routine to verify that the map (2.2) sends H + n to End n H + and themap (2.3) sends H − n to End − n H + for all n ∈ N . Definition 2.6 (The Heisenberg double, [STS94, Def. 3.1]) . We define h ( H + , H − ) to bethe Heisenberg double of H + . More precisely h ( H + , H − ) ∼ = H + ⊗ H − as k -modules, and wewrite a x for a ⊗ x , a ∈ H + , x ∈ H − , viewed as an element of h ( H + , H − ). Multiplicationis given by(2.4) ( a x )( b y ) := P ( x ) a R x (1) ∗ ( b ) x (2) y = P ( x ) , ( b ) h x (1) , b (2) i ab (1) x (2) y. We will often view H + and H − as subalgebras of h ( H + , H − ) via the maps a a x x for a ∈ H + and x ∈ H − . Then we have ax = a x . When the context is clear,we will simply write h for h ( H + , H − ). We have a natural grading h = L n ∈ Z h n , where h n = L k − ℓ = n H + k H − ℓ . Remark 2.7.
The Heisenberg double is a twist of the Drinfeld quantum double by a right2-cocycle (see [Lu94, Th. 6.2]).
Lemma 2.8. If x ∈ H − and a, b ∈ H + , then R x ∗ ( ab ) = P ( x ) R x (1) ∗ ( a ) R x (2) ∗ ( b ) . Proof.
For x, y ∈ H − and a, b ∈ H + , we have h y, R x ∗ ( ab ) i = h yx, ab i = h ∆( yx ) , a ⊗ b i = h ∆( y )∆( x ) , a ⊗ b i = h ∆( y ) , R ∆( x ) ∗ ( a ⊗ b ) i = h ∆( y ) , P ( x ) R x ∗ (1) ( a ) ⊗ R x ∗ (2) ( b ) i = h y, P ( x ) R x ∗ (1) ( a ) R x ∗ (2) ( b ) i . The result then follows from the nondegeneracy of the bilinear form. (cid:3)
Interchanging H − and H + in the construction of the Heisenberg double results in theopposite algebra: h ( H − , H + ) ∼ = h ( H + , H − ) op (see [Lu94, Prop. 5.3]).2.3. Fock space.
We now introduce a natural representation of the algebra h . Definition 2.9 (Vacuum vector) . An element v of an h -module V is called a lowest weight (resp. highest weight ) vacuum vector if k v ∼ = k and H − v = 0 (resp. H + v = 0). Definition 2.10 (Fock space) . The algebra h has a natural (left) representation on H + given by ( a x )( b ) = a R x ∗ ( b ) , a, b ∈ H + , x ∈ H − . We call this the lowest weight Fock space representation of h ( H + , H − ) and denote it by F = F ( H + , H − ). Note that this representation is generated by the lowest weight vacuumvector 1 ∈ H + .Suppose X + is a subalgebra of H + that is invariant under the left-regular action of H − on H + . (Note that it follows that X + is a graded subalgebra of H + .) Then X + H − is asubalgebra of h ( H + , H − ) acting naturally on X + . The following result (when X + = H + ) is ageneralization of the Stone–von Neumann Theorem to the setting of an arbitrary Heisenbergdouble. ATEGORIFICATION AND HEISENBERG DOUBLES ARISING FROM TOWERS OF ALGEBRAS 7
Theorem 2.11.
Let X + be a subalgebra of H + that is invariant under the left-regular actionof H − on H + .(a) The only ( X + H − ) -submodules of X + are those of the form IX + for some ideal I of k . In particular, if k is a field, then X + is irreducible as an ( X + H − ) -module.(b) Let k − ∼ = k (isomorphism of k -modules) be the representation of H − such that H − n acts as zero for all n > and H − ∼ = k acts by left multiplication. Then X + isisomorphic to the induced module Ind X + H − H − k − as an ( X + H − ) -module.(c) Any ( X + H − ) -module generated by a lowest weight vacuum vector is isomorphic to X + .If X + = H + then X + H − = h ( H + , H − ) and the module X + is the lowest weight Fock space F . In that case we also have the following.(d) The lowest weight Fock space representation F of h is faithful.Proof. Throughout the proof, we write 1 for 1 H + = 1 X + .(a) Clearly, if I is an ideal of k , then IX + is a submodule of X + . Now suppose W ⊆ X + is a nonzero submodule, and let I = { c ∈ k | c ∈ W } . It is easy to see that I is an ideal of k . We claim that W = IX + . Since the element1 generates X + , we clearly have IX + ⊆ W . Now suppose there exists a ∈ W such that a IX + . Without loss of generality, we can write a = P ℓn =0 a n for a n ∈ H + n , ℓ ∈ N , and a ℓ IX + (otherwise, consider a − a ℓ ). Let b , . . . , b m be a basis of H + ℓ such that b , . . . , b k is a basis of X + ℓ , for k = dim k X + ℓ . Let x , . . . , x m be the dual basis of H − ℓ . Then it is easyto verify that P kj =1 b j x j acts as the identity on X + ℓ and as zero on X + n for n < ℓ . Thus, a ℓ = P kj =1 ( b j x j )( a ) ∈ P kj =1 b j IX + ⊆ IX + , since R x j ∗ ( a ) ∈ W ∩ H +0 for all j = 1 , . . . , k . This contradiction completes the proof.(b) We have an injective homomorphism of H − -modules k − ֒ → Res X + H − H − X + , . Since induction is left adjoint to restriction (see, for example, [CR81, (2.19)]), this gives riseto a homomorphism of h -modules(2.5) Ind X + H − H − k − → X + , . Since the element 1 generates X + , this map is surjective. Now,Ind X + H − H − k − = ( X + H − ) ⊗ H − k − = X + H − ⊗ H − k − . It follows that, as a left X + -module, Ind X + H − H − k − is a quotient of X + ⊗ k k − ∼ = X + and themap (2.5) is the identity map, hence an isomorphism.(c) Suppose V is a representation of X + H − generated by a lowest weight vacuum vector v . Then, as above, we have an injective homomorphism of H − -modules k − ֒ → Res X + H − H − V, v , and thus a homomorphism of ( X + H − )-modules(2.6) Ind X + H − H − k − → V, v . ALISTAIR SAVAGE AND ODED YACOBI
Since V is generated by v , this map is surjective. Since Ind X + H − H − k − ∼ = X + , it followseasily from part (a) that it is also injective.(d) Suppose α is a nonzero element of h . Write α = α ′ + α ′′ where α ′ is a nonzero elementof H + H − n for some n ∈ N and α ′′ ∈ P k>n H + H − k . Choose a basis x , . . . , x m of H − n andlet b , . . . , b m denote the dual basis of H + n . Then we can write α ′ = P mj =1 a j x j for some a j ∈ H + . Since α ′ = 0, we have a j = 0 for some j . Then α ( b j ) = α ′ ( b j ) = a j = 0. Thus theaction of h on F is faithful. (cid:3) Remark 2.12.
By Theorem 2.11(d), we may view h as the subalgebra of End H + generatedby L a , a ∈ H + , and R x ∗ , x ∈ H − .3. Towers of algebras and categorification of Fock space
In this section, we consider dual Hopf algebras arising from towers of algebras. In thiscase, we are able to deduce some further results about the Heisenberg double h . We thenprove our main result, that towers of algebras give rise to categorifications of the lowestweight Fock space representation of h (Theorem 3.18). Recall that F is an arbitrary field.3.1. Modules categories and their Grothendieck groups.
For an arbitrary F -algebra B , let B -mod denote the category of finitely generated left B -modules and let B -pmoddenote the category of finitely generated projective left B -modules. We then define G ( B ) = K ( B -mod) and K ( B ) = K ( B -pmod) , where K ( B -mod) denotes the Grothendieck group of the abelian category B -mod, and K ( B -pmod) denotes the (split) Grothendieck group of the additive category B -pmod. For C = B -mod or B -pmod, we denote the class of an object M ∈ C in K ( C ) by [ M ].There is a natural bilinear form(3.1) h· , ·i : K ( B ) ⊗ G ( B ) → Z , h [ P ] , [ M ] i = dim F Hom B ( P, M ) . If B is a finite dimensional algebra, let V , . . . , V s be a complete list of nonisomorphic simple B -modules. If P i is the projective cover of V i for i = 1 , . . . , s , then P , . . . , P s is a com-plete list of nonisomorphic indecomposable projective B -modules (see, for example, [ARS95,Cor. I.4.5]) and we have G ( B ) = L si =1 Z [ V i ] and K ( B ) = L si =1 Z [ P i ] . If F is algebraically closed, then(3.2) h [ P i ] , [ V j ] i = δ ij for 1 ≤ i, j ≤ s, and so the pairing (3.1) is perfect.Suppose ϕ : B → A is an algebra homomorphism. Then we can consider A as a left B -module via the action ( b, a ) ϕ ( b ) a . Similarly, we can consider A as a right B -module.Then we have induction and restriction functorsInd AB : B -mod → A -mod , Ind AB N := A ⊗ B N, N ∈ B -mod , Res AB : A -mod → B -mod , Res AB M := Hom A ( A, M ) ∼ = B A ⊗ A M, M ∈ A -mod , where B A denotes A considered as a ( B, A )-bimodule and the left B -action on Hom A ( A, M ) isgiven by ( b, f ) f ◦ R b for f ∈ Hom A ( A, M ) and b ∈ B (here R b denotes right multiplication ATEGORIFICATION AND HEISENBERG DOUBLES ARISING FROM TOWERS OF ALGEBRAS 9 by b ). The isomorphism above is given by the map f ⊗ f (1 A ) for f ∈ Hom A ( A, M ). Thisisomorphism is natural in M and so we have an isomorphism of functors Res AB ∼ = B A ⊗ A − .3.2. Towers of algebras.Definition 3.1 (Tower of algebras) . Let A = L n ∈ N A n be a graded algebra over a field F with multiplication ρ : A ⊗ A → A . Then A is called a tower of algebras if the followingconditions are satisfied:(TA1) Each graded piece A n , n ∈ N , is a finite dimensional algebra (with a different multi-plication) with a unit 1 n . We have A ∼ = F .(TA2) The external multiplication ρ m,n : A m ⊗ A n → A m + n is a homomorphism of algebrasfor all m, n ∈ N (sending 1 m ⊗ n to 1 m + n ).(TA3) We have that A m + n is a two-sided projective ( A m ⊗ A n )-module with the actiondefined by a · ( b ⊗ c ) = aρ m,n ( b ⊗ c ) and ( b ⊗ c ) · a = ρ m,n ( b ⊗ c ) a, for all m, n ∈ N , a ∈ A m + n , b ∈ A m , c ∈ A n .(TA4) For each n ∈ N , the pairing (3.1) (with B = A n ) is perfect. (Note that this conditionis automatically satisfied if F is an algebraically closed field, by (3.2).)For the remainder of this section we assume that A is a tower of algebras. We let(3.3) G ( A ) = L n ∈ N G ( A n ) and K ( A ) = L n ∈ N K ( A n ) . Then we have a perfect pairing h· , ·i : K ( A ) × G ( A ) → Z given by(3.4) h [ P ] , [ M ] i = ( dim F (Hom A n ( P, M )) if P ∈ A n -pmod and M ∈ A n -mod for some n ∈ N , . We also define a perfect pairing h· , ·i : ( K ( A ) ⊗ K ( A )) × ( G ( A ) ⊗ G ( A )) by h [ P ] ⊗ [ Q ] , [ M ] ⊗ [ N ] i = dim F (Hom A k ⊗ A ℓ ( P ⊗ Q, M ⊗ N )) if P ∈ A k -pmod , Q ∈ A ℓ -pmodand M ∈ A k -mod , N ∈ A ℓ -mod for some k, ℓ ∈ N , . Thus we have h [ P ] ⊗ [ Q ] , [ M ] ⊗ [ N ] i = h [ P ] , [ M ] ih [ Q ] , [ N ] i .Consider the direct sums of categories A -mod := L n ∈ N A n -mod , A -pmod := L n ∈ N A n -pmod . For r ∈ N + , we define A -mod ⊗ r := L n ,...,n r ∈ N ( A n ⊗ · · · ⊗ A n r ) -mod ,A -pmod ⊗ r := L n ,...,n r ∈ N ( A n ⊗ · · · ⊗ A n r ) -pmod . Then, for i, j ∈ { , . . . , r } , i < j , we define S ij : A -mod ⊗ r → A -mod ⊗ r to be the endo-functor that interchanges the i th and j th factors, that is, the endofunctor arising from theisomorphism A n ⊗ · · · ⊗ A n r ∼ = A n ⊗ · · · ⊗ A n i − ⊗ A n j ⊗ A n i +1 ⊗ · · · ⊗ A n j − ⊗ A n i ⊗ A n j +1 ⊗ · · · ⊗ A n r . We use the same notation to denote the analogous endofunctor on A -pmod ⊗ r . We also have the following functors: ∇ : A -mod ⊗ → A -mod , ∇| ( A m ⊗ A n ) -mod = Ind A m + n A m ⊗ A n , ∆ : A -mod → A -mod ⊗ , ∆ | A n -mod = L k + ℓ = n Res A n A k ⊗ A ℓ ,η : Vect → A -mod , η ( V ) = V ∈ A -mod for V ∈ Vect ,ε : A -mod → Vect , ε ( V ) = ( V if V ∈ A -mod , . (3.5)In the above, we have identified A -mod with the category Vect of finite dimensional vectorspaces over F . Replacing A -mod by A -pmod above, we also have the functors ∇ , ∆, η and ε on A -pmod. Since the above functors are all exact (we use axiom (TA3) here), they inducea multiplication, comultiplication, unit and counit on G ( A ) and K ( A ). We use the samenotation to denote these induced maps.Since induction is always left adjoint to restriction (see, for example, [CR81, (2.19)]), ∇ isleft adjoint to ∆. However, in many examples of towers of algebras (e.g. nilcoxeter algebrasand 0-Hecke algebras), induction is not right adjoint to restriction. Nevertheless, we oftenhave something quite close to this property. Any algebra automorphism ψ n of A n inducesan isomorphism of categories Ψ n : A n -mod → A n -mod (which restricts to an isomorphismof categories Ψ n : A n -pmod → A n -pmod) by twisting the A n action. Then Ψ := L n ∈ N Ψ n isan automorphism of the categories A -mod and A -pmod. It induces automorphisms (whichwe also denote by Ψ) of G ( A ) and K ( A ). Definition 3.2 (Conjugate adjointness) . Given a tower of algebras A , we say that inductionis conjugate right adjoint to restriction (and restriction is conjugate left adjoint to induction)with conjugation Ψ if there are isomorphisms of algebras ψ n : A n → A n , n ∈ N , such that ∇ is right adjoint to Ψ ⊗ ∆Ψ − .The following lemma will be useful since many examples of towers of algebras are in factcomposed of Frobenius algebras. We refer the reader to [SY11] for background on Frobeniusalgebras. Lemma 3.3.
If each A n , n ∈ N , is a Frobenius algebra, then induction is conjugate rightadjoint to restriction, with conjugation given by ψ n being the inverse of the Nakayama auto-morphism of A n .Proof. This follows from [Kho01, Lem. 1] by taking B = A m + n , B = A m ⊗ A n and N tobe A m + n , considered as an ( A m + n , A m ⊗ A n )-bimodule in the natural way. (cid:3) Definition 3.4 (Strong tower of algebras) . We say that a tower of algebras A is strong ifinduction is conjugate right adjoint to restriction and we have an isomorphism of functors(3.6) ∆ ∇ ∼ = ∇ ⊗ S ∆ ⊗ . Remark 3.5.
The isomorphism (3.6) is a compatibility between induction and restrictionthat is analogous to the well known Mackey theory for finite groups. It implies that K ( A ) and G ( A ) are Hopf algebras under the operations defined above (see [BL09, Th. 3.5] – althoughthe authors of that paper work over C and they assume that the external multiplicationmaps ρ m,n are injective, the arguments hold in the more general setting considered here). ATEGORIFICATION AND HEISENBERG DOUBLES ARISING FROM TOWERS OF ALGEBRAS 11
Definition 3.6 (Dualizing tower of algebras) . We say that a tower of algebras A is dualizing if, under the operations defined above, K ( A ) and G ( A ) are Hopf algebras which are dual,in the sense of Definition 2.4 (with k = Z ), under the bilinear form (3.4) (i.e. the perfectpairing (3.4) is a Hopf pairing). Proposition 3.7.
Suppose A is a strong tower of algebras with conjugation Ψ . Then thefollowing statements are equivalent.(a) The tower A is dualizing.(b) We have Ψ ⊗ ∆Ψ − ( P ) ∼ = ∆( P ) for all P ∈ A -pmod .(c) We have Ψ ⊗ ∆Ψ − = ∆ as maps K ( A ) → K ( A ) ⊗ K ( A ) .In particular A is dualizing if each A n is a symmetric algebra (i.e. if Ψ = Id ) or, moregenerally, if Ψ acts trivially on K ( A ) .Proof. First assume that (b) holds. With one exception, the proof that A is dualizing thenproceeds exactly as in the proof of [BL09, Th. 3.6] since (3.6) implies axiom (5) in [BL09, § C and they assume that the exter-nal multiplication maps ρ m,n are injective, the arguments hold in the more general settingconsidered here.) The one exception is in the proof that(3.7) h ∆([ P ]) , [ M ] ⊗ [ N ] i = h [ P ] , ∇ ([ M ] ⊗ [ N ]) i , for all M ∈ A m -mod, N ∈ A n -mod, P ∈ A m + n -pmod. However, under our assumptions,we haveHom A m + n ( P, ∇ ( M ⊗ N )) ∼ = Hom A m ⊗ A n (Ψ ⊗ ∆Ψ − ( P ) , M ⊗ N ) ∼ = Hom A m ⊗ A n (∆( P ) , M ⊗ N ) , which immediately implies (3.7). Thus A is dualizing.Now suppose A is dualizing. Then, for all P ∈ A -pmod and M ∈ A -mod ⊗ , we have h Ψ ⊗ ∆Ψ − ([ P ]) , [ M ] i = h [ P ] , ∇ ([ M ]) i = h ∆([ P ]) , [ M ] i , where the first equality holds by the assumption that induction is conjugate right adjointto restriction and the second equality holds by the assumption that the tower is dualizing.Then (c) follows from the nondegeneracy of the bilinear form.The fact that (b) and (c) are equivalent follows from the fact that every short exactsequence of projective modules splits. Thus, for P, Q ∈ A -pmod, we have [ P ] = [ Q ] in K ( A )if and only if P ∼ = Q . (cid:3) Remark 3.8.
It is crucial in (b) that P be a projective module. The isomorphism does nothold, in general, for arbitrary modules, even if the tower is dualizing. For instance, the towerof 0-Hecke algebras is dualizing (see Corollary 4.6), but one can show that Ψ ⊗ ∆Ψ − ∼ = S ∆(see Lemma 4.4). Then (b) corresponds to the fact that the comultiplication on NSym iscocommutative. However, the comultiplication on QSym is not cocommutative and thus (b)does not hold, in general, if P is not projective. We refer the reader to Section 8 for furtherdetails on the tower of 0-Hecke algebras.3.3. The Heisenberg double associated to a tower of algebras.
In this section weapply the constructions of Section 2 to the dual pair ( G ( A ) , K ( A )) arising from a dualizingtower of algebras A . We also see that some natural subalgebras of the Heisenberg doublearise in this situation. Definition 3.9 ( h ( A ), F ( A ), G proj ( A )) . Suppose A is a dualizing tower of algebras. We let h ( A ) = h ( G ( A ) , K ( A )) and F ( A ) = F ( G ( A ) , K ( A )). For each n ∈ N , A n -pmod is a fullsubcategory of A n -mod. The inclusion functor induces the Cartan map K ( A ) → G ( A ). Let G proj ( A ) denote the image of the Cartan map.For the remainder of this section, we fix a dualizing tower of algebras A and let(3.8) H − = K ( A ) , H + = G ( A ) , H +proj = G proj ( A ) , h = h ( A ) , F = F ( A ) . To avoid confusion between G ( A ) and K ( A ), we will write [ M ] + to denote the class of afinitely generated (possibly projective) A n -module in H + and [ M ] − to denote the class of afinitely generated projective A n -module in H − . If P ∈ A p -pmod and N ∈ ( A n − p ⊗ A p ) -mod,then we have a natural A n − p -module structure on Hom A p ( P, N ) given by(3.9) ( a · f )( b ) = ( a ⊗ · ( f ( b )) , a ∈ A n − p , f ∈ Hom A p ( P, N ) , b ∈ P. Lemma 3.10. If p, n ∈ N , P ∈ A p -pmod and N ∈ A n -mod , then we have [ P ] − · [ N ] + = ( if p > n, [Hom A p ( P, Res A n A n − p ⊗ A p N )] + if p ≤ n. Here · denotes the action of h on F and Hom A p ( P, Res A n A n − p ⊗ A p N ) is viewed as an A n − p -module as in (3.9) .Proof. The case p > n follows immediately from the fact that H + n − p = 0 if p > n . Assume p ≤ n . For R ∈ A n − p -pmod, we have h [ R ] − , [ P ] − · [ N ] + i = h [ R ] − [ P ] − , [ N ] + i = h∇ ([ R ] − ⊗ [ P ] − ) , [ N ] + i = h [ R ] − ⊗ [ P ] − , ∆([ N ] + ) i = dim F Hom A n − p ⊗ A p ( R ⊗ P, Res A n A n − p ⊗ A p N )= dim F Hom A n − p ( R, Hom A p ( P, Res A n A n − p ⊗ A p N ))= h [ R ] − , [Hom A p ( P, Res A n A n − p ⊗ A p N )] + i . The result then follows from the nondegeneracy of the bilinear form. (cid:3)
Lemma 3.11.
Suppose k is a commutative ring and R and S are k -algebras. Furthermore,suppose that P is a finitely generated projective S -module and Q is a finitely generatedprojective ( R ⊗ S ) -module. Then Hom S ( P, Q ) is a finitely generated projective R -module.Proof. If P is a projective S -module and Q is a projective ( R ⊗ S )-module then there exist s, t ∈ N , an S -module P ′ and an ( R ⊗ S )-module Q ′ such that P ⊕ P ′ ∼ = S s and Q ⊕ Q ′ ∼ =( R ⊗ S ) t . Then we haveHom S ( P, Q ) ⊕ Hom S ( P, Q ′ ) ⊕ Hom S ( P ′ , ( R ⊗ S ) t ) ∼ = Hom S ( S s , ( R ⊗ S ) t ) ∼ = Hom S ( S, R ⊗ S ) st ∼ = R st . Thus Hom S ( P, Q ) is a projective R -module. (cid:3) Proposition 3.12.
We have that H +proj is a subalgebra of H + that is invariant under theleft-regular action of H − . ATEGORIFICATION AND HEISENBERG DOUBLES ARISING FROM TOWERS OF ALGEBRAS 13
Proof.
Assume P ∈ A p -pmod and Q ∈ A q -pmod. As in the proof of [BL09, Prop. 3.2], wehave that Ind A p + q A p ⊗ A q ( P ⊗ Q ) is a projective A p + q -module. It follows that H +proj is a subalgebraof H + .By Lemma 3.10, it remains to show thatHom A p ( P, Res A q A q − p ⊗ A p Q ) ∈ A q − p -pmod . Again, as in the proof of [BL09, Prop. 3.2], we have that Res A q A q − p ⊗ A p Q is a projective( A q − p ⊗ A p )-module. The result then follows from Lemma 3.11. (cid:3) Definition 3.13 (The projective Heisenberg double h proj ( A )) . By Proposition 3.12, h proj = h proj ( A ) := H +proj H − is a subalgebra of h . In other words, h proj is the subalgebra of h generated by H +proj and H − (viewing the latter two as Z -submodules of h as in Definition 2.6).We call h proj the projective Heisenberg double associated to A . Definition 3.14 (Fock space F proj ( A ) of h proj ) . By Proposition 3.12, the algebra h proj actson H +proj . We call this the lowest weight Fock space representation of h proj and denote it by F proj = F proj ( A ). Note that this representation is generated by the lowest weight vacuumvector 1 ∈ H +proj . Proposition 3.15.
The Fock space F proj of h proj has the following properties.(a) The only submodules of F proj are those submodules of the form n F proj for n ∈ Z .(b) Let Z − ∼ = Z (isomorphism of Z -modules) be the representation of H − such that H − n acts as zero for all n > and H − ∼ = Z acts by left multiplication. Then F proj isisomorphic to the induced module Ind h proj H − Z − as an h proj -module.(c) Any representation of h proj generated by a lowest weight vacuum vector is isomorphicto F proj .Proof. This is an immediate consequence of Theorem 2.11, taking X + = H +proj . (cid:3) Note that the lowest weight Fock space F proj is not a faithful h proj -module in general (seeSection 8.5), in contrast to the case for h (see Theorem 2.11(d)). However, we can definea highest weight Fock space of h proj that is faithful. Consider the augmentation algebrahomomorphism ǫ + : H +proj → Z uniquely determined by ǫ + ( H + n ∩ H +proj ) = 0 for n >
0. Let Z + ǫ denote the corresponding H +proj -module. We call the induced module F − proj := Ind h proj H +proj Z + ǫ the highest weight Fock space representation of h proj . It is generated by the highest weightvacuum vector 1 ∈ Z + ǫ . Proposition 3.16.
The highest weight Fock space representation F − proj of h proj is faithful.Proof. We have h proj ∼ = H +proj ⊗ H − as k -modules and so F − proj ∼ = H − as k -modules. Theaction of h proj on F − proj is simply the restriction of the natural action of h on H − , which isfaithful by (an obvious highest weight analogue of) Theorem 2.11(d). (cid:3) Categorification of Fock space.
In this section we prove our main result, the cate-gorification of the Fock space representation of the Heisenberg double. We continue to fix adualizing tower of algebras A and to use the notation of (3.8). Recall the direct sums of categories A -mod = L n ∈ N A n -mod , A -pmod = L n ∈ N A n -pmod . For each M ∈ A m -mod, m ∈ N , define the functor Ind M : A -mod → A -mod byInd M ( N ) = Ind A m + n A m ⊗ A n ( M ⊗ N ) ∈ A m + n -mod , N ∈ A n -mod , n ∈ N . For each P ∈ A p -pmod, p ∈ N , define the functor Res P : A -mod → A -mod byRes P ( N ) = Hom A p ( P, Res A n A n − p ⊗ A p N ) ∈ A n − p -mod , N ∈ A n -mod , n ∈ N , where Res P ( N ) is interpreted to be the zero object of A -mod if n − p < P ( A -pmod) ⊆ A -pmod , Res P ( A -pmod) ⊆ A -pmod for all P ∈ A -pmod . Thus we have the induced functors Ind P , Res P : A -pmod → A -pmod for P ∈ A -pmod.Since the functors Ind M and Res P are exact for all M ∈ A -mod and P ∈ A -pmod,they induce endomorphisms [Ind M ] and [Res P ] of G ( A ). Similarly, Ind P and Res P induceendomorphisms [Ind P ] and [Res P ] of G proj ( A ) for all P ∈ A -pmod. Proposition 3.17.
Suppose A is a dualizing tower of algebras.(a) For all M, N ∈ A -mod and P ∈ A -pmod , we have ([ M ] P ])([ N ]) = [Ind M ] ◦ [Res P ]([ N ]) = [Ind M ◦ Res P ( N )] ∈ G ( A ) . (b) For all Q, P, R ∈ A -pmod , we have ([ Q ] P ])([ R ]) = [Ind Q ] ◦ [Res P ]([ R ]) = [Ind Q ◦ Res P ( R )] ∈ G proj ( A ) . Proof.
This follows from the definition of the multiplication in G ( A ) and Lemma 3.10. (cid:3) Part (a) (resp. part (b)) of Proposition 3.17 shows how the action of h on F (resp. h proj on F proj ) is induced by functors on L n ≥ A n -mod (resp. L n ≥ A n -pmod). Typically a categorification of a representation consists of isomorphisms of such functors which lift thealgebra relations. As we now describe, this can be done if the tower of algebras is strong.First note that the algebra structure on h is uniquely determined by the fact that H ± aresubalgebras and by the relation(3.10) xa = P ( x ) R x (1) ∗ ( a ) x (2) , x ∈ H − , a ∈ H + , between H − and H + . Since the natural action of h on H + is faithful by Theorem 2.11(d),equation (3.10) is equivalent to the equalities in End H + (3.11) R x ∗ ◦ L a = ∇ (cid:16) R ∆( x ) ∗ ( a ⊗ − ) (cid:17) , x ∈ H − , a ∈ H + . For Q ∈ ( A p ⊗ A q ) -pmod, M ∈ A m -mod, and N ∈ A n -mod, defineRes Q ( M ⊗ N ) := Hom A p ⊗ A q (cid:16) Q, S (cid:16) Res A m A m − p ⊗ A p M ⊗ Res A n A n − q ⊗ A q N (cid:17)(cid:17) . For Q ∈ A -mod ⊗ , we let Res Q denote the corresponding sum of functors. ATEGORIFICATION AND HEISENBERG DOUBLES ARISING FROM TOWERS OF ALGEBRAS 15
Theorem 3.18.
Suppose that A is a strong tower of algebras. Then we have the followingisomorphisms of functors for all M, N ∈ A -mod and P, Q ∈ A -pmod . Ind M ◦ Ind N ∼ = Ind ∇ ( M ⊗ N ) , (3.12) Res P ◦ Res Q ∼ = Res ∇ ( P ⊗ Q ) , (3.13) Res P ◦ Ind M ∼ = ∇ Res Ψ ⊗ ∆Ψ − ( P ) ( M ⊗ − ) . (3.14) In particular, if A is dualizing, then the above yields a categorification of the lowest weightFock space representations of h ( A ) and h proj ( A ) . The isomorphisms (3.12) and (3.13) categorify the multiplication in G ( A ) and K ( A ) re-spectively. If A is dualizing, then, in light of Proposition 3.7, the isomorphism (3.14) cat-egorifies the relation (3.11). Thus, Theorem 3.18 provides a categorification of the lowestweight Fock space representation F ( A ) of h ( A ). If we restrict the induction and restrictionfunctors to A -pmod and require M, N ∈ A -pmod in the statement of the theorem, then weobtain a categorification of the lowest weight Fock space representation F proj ( A ) of h proj ( A ).Note that the categorification in Theorem 3.18 does not rely on a particular presentationof the Heisenberg double h ( A ) (see Remark 6.1), in contrast to many other categorificationstatements appearing in the literature. Proof of Theorem 3.18.
Suppose M ∈ A m -mod, N ∈ A n -mod, P ∈ A p -pmod, Q ∈ A q -pmod,and L ∈ A ℓ -mod. Then we haveInd M ◦ Ind N ( L ) = Ind A m + n + ℓ A m ⊗ A n + ℓ (cid:16) M ⊗ Ind A n + ℓ A n ⊗ A ℓ ( N ⊗ L ) (cid:17) ∼ = Ind A m + n + ℓ A m ⊗ A n + ℓ Ind A m ⊗ A n + ℓ A m ⊗ A n ⊗ A ℓ ( M ⊗ N ⊗ L ) ∼ = Ind A m + n + ℓ A m ⊗ A n ⊗ A ℓ ( M ⊗ N ⊗ L ) ∼ = Ind A m + n + ℓ A m + n ⊗ A ℓ Ind A m + n ⊗ A ℓ A m ⊗ A n ⊗ A ℓ ( M ⊗ N ⊗ L ) ∼ = Ind A m + n + ℓ A m + n ⊗ A ℓ (cid:16) Ind A m + n A m ⊗ A n ( M ⊗ N ) ⊗ L (cid:17) ∼ = Ind ∇ ( M ⊗ N ) L. Since each of the above isomorphisms is natural in L , this proves (3.12).Similarly, we haveRes P ◦ Res Q ( L ) = Hom A p ( P, Res A ℓ − q A ℓ − q − p ⊗ A p Hom A q ( Q, Res A ℓ A ℓ − q ⊗ A q L )) ∼ = Hom A p ( P, Hom A q ( Q, Res A ℓ A ℓ − p − q ⊗ A p ⊗ A q L )) ∼ = Hom A p ⊗ A q ( P ⊗ Q, Res A ℓ A ℓ − p − q ⊗ A p ⊗ A q L ) ∼ = Hom A p ⊗ A q ( P ⊗ Q, Res A ℓ − p − q ⊗ A p + q A ℓ − p − q ⊗ A p ⊗ A q Res A ℓ A ℓ − p − q ⊗ A p + q L ) ∼ = Hom A p + q (Ind A p + q A p ⊗ A q ( P ⊗ Q ) , Res A ℓ A ℓ − p − q ⊗ A p + q L ) ∼ = Res ∇ ( P ⊗ Q ) L. Since each of the above isomorphisms is natural in L , this proves (3.13).Finally, we haveRes P ◦ Ind M ( L ) = Hom A p ( P, Res A m + ℓ A m + ℓ − p ⊗ A p Ind A m + ℓ A m ⊗ A ℓ ( M ⊗ L )) ∼ = Hom A p ( P, L s + t = p Ind A m + ℓ − p ⊗ A p A m − s ⊗ A s ⊗ A ℓ − t ⊗ A t Res A m ⊗ A ℓ A m − s ⊗ A s ⊗ A ℓ − t ⊗ A t ( M ⊗ L )) ∼ = Hom A p ( P, L s + t = p Ind A m + ℓ − p ⊗ A p A m − s ⊗ A ℓ − t ⊗ A p Ind A m − s ⊗ A ℓ − t ⊗ A p A m − s ⊗ A s ⊗ A ℓ − t ⊗ A t Res A m ⊗ A ℓ A m − s ⊗ A s ⊗ A ℓ − t ⊗ A t ( M ⊗ L )) ∼ = L s + t = p Ind A m + ℓ − p A m − s ⊗ A ℓ − t Hom A p ( P, Ind A m − s ⊗ A ℓ − t ⊗ A p A m − s ⊗ A s ⊗ A ℓ − t ⊗ A t Res A m ⊗ A ℓ A m − s ⊗ A s ⊗ A ℓ − t ⊗ A t ( M ⊗ L )) ∼ = L s + t = p Ind A m + ℓ − p A m − s ⊗ A ℓ − t Hom A s ⊗ A t (Ψ ⊗ Res A p A s ⊗ A t Ψ − ( P ) , Res A m ⊗ A ℓ A m − s ⊗ A s ⊗ A ℓ − t ⊗ A t ( M ⊗ L )) ∼ = ∇ Res Ψ ⊗ ∆Ψ − ( P ) ( M ⊗ L ) , where the first isomorphism follows from (3.6). Since all of the above isomorphisms arenatural in L , this proves (3.14).The final assertion of the theorem follows from Proposition 3.7 as explained in the para-graph following the statement of the theorem. (cid:3) Hecke-like towers of algebras
In the remainder of the paper we will be studying some well known examples of towersof algebras. These examples are all quotients of groups algebras of braid groups by qua-dratic relations. In this subsection, we prove that all such towers of algebras are strong anddualizing. In this section, F is an algebraically closed field unless otherwise specified. Definition 4.1 (Hecke-like algebras and towers) . We say that the F -algebra B is Hecke-like (of degree n ) if it is generated by elements T , . . . , T n − , subject to the relations T i T j = T j T i , | i − j | > ,T i T i +1 T i = T i +1 T i T i +1 , i = 1 , . . . , n − ,T i = cT i + d, i = 1 , . . . , n − , for some c, d ∈ F (independent of i ). In other words, B is Hecke-like if it is a quotient of thegroup algebra of the braid group by quadratic relations (the last set of relations above).If, for n ∈ N , the algebra A n is a Hecke-like algebra of degree n , and the constants c, d above are independent of n , then we can define an external multiplication on A = L n ∈ N A n by(4.1) ρ m,n : A m ⊗ A n → A m + n , T i ⊗ T i , ⊗ T i T m + i . Axioms (TA1) and (TA2) follow immediately. Furthermore, it follows from Lemma 4.2 belowthat, as a left ( A m ⊗ A n )-module, we have A m + n = L w ∈ X m,n ( A m ⊗ A n ) T w , where X m,n is a set of minimal length representatives of the cosets ( S m × S n ) \ S m + n . Thus A m + n is a projective left ( A m ⊗ A n )-module. Similarly, it is also a projective right ( A m ⊗ A n )-module and so axiom (TA3) is satisfied. Finally (TA4) is satisfied since F is algebraicallyclosed. We call the resulting tower of algebras a Hecke-like tower of algebras . Lemma 4.2.
Suppose that B is a Hecke-like algebra of degree n .(a) If, for w ∈ S n , we define T w = T i · · · T i r , where s i · · · s i r is a reduced decompositionof w (these elements are well defined by the braid relations), then { T w | w ∈ S n } is abasis of B . In particular, the dimension of B is n ! . ATEGORIFICATION AND HEISENBERG DOUBLES ARISING FROM TOWERS OF ALGEBRAS 17 (b) We have T w T w = T w w for all w , w ∈ S n such that ℓ ( w w ) = ℓ ( w ) + ℓ ( w ) , where ℓ is the usual length function on S n .(c) The algebra B is a Frobenius algebra with trace map λ : B → F given by λ ( T w ) = δ w,w ,where w is the longest element of S n . The corresponding Nakayama automorphismis the map ψ n : B → B given by ψ n ( T i ) = T n − i .Proof. The proof of parts (a) and (b) is analogous to the proof for the usual Hecke algebraof type A and is left to the reader. It remains to prove part (c).Suppose B is a Hecke-like algebra of degree n . To show that B is a Frobenius algebrawith trace map λ , it suffices to show that ker λ contains no nonzero left ideals. Let I be anonzero ideal of B . Then choose a nonzero element b = P w ∈ S n a w T w ∈ I and let τ be amaximal length element of the set { w ∈ S n | a w = 0 } . Then we have λ ( T w τ − b ) = a τ = 0.Thus I is not contained in ker λ .To show that ψ n is the Nakayama automorphism, it suffices to show that λ ( T w T i ) = λ ( T n − i T w ) for all i ∈ { , . . . , n − } and w ∈ S n . We break the proof into four cases. Case 1: ℓ ( w ) ≤ ℓ ( w ) −
2. In this case we clearly have λ ( T w T i ) = 0 = λ ( T n − i T w ). Case 2: w = w . Then we can write w = w = τ s i for some τ ∈ S n with ℓ ( τ ) = ℓ ( w ) − λ ( T w T i ) = λ ( cT w + dT τ ) = c . Now, since w s i = s n − i w , we have w = w = s n − i τ .Thus λ ( T n − i T w ) = λ ( cT w + dT τ ) = c = λ ( T w T i ). Case 3: ws i = w . Since, as noted above, we have w s i = s n − i w , it follows that s n − i w = w and so λ ( T w T i ) = λ ( T w ) = λ ( T n − i T w ). Case 4: ℓ ( w ) = ℓ ( w ) −
1, but ws i = w . Then we have w = τ s i for some τ ∈ S n with ℓ ( τ ) = ℓ ( w ) −
1. Thus λ ( T w T i ) = λ ( cT w + dT τ ) = 0. Using again the equality w s i = s n − i w ,we have s n − i w = w . Then an analogous argument shows that λ ( T n − i T w ) = 0. (cid:3) When considering a Hecke-like tower of algebras, we will always assume that the conjuga-tion Ψ is given by the Nakayama automorphisms ψ n described above (see Proposition 3.3). Proposition 4.3.
All Hecke-like towers of algebras are strong.Proof.
Suppose A = L n ∈ N A n is a Hecke-like tower of algebras. We formulate the isomor-phism (3.6) in terms of bimodules. Fix n, m, k, ℓ such that n + m = k + ℓ and set N = n + m .Let ( k,ℓ ) ( A N ) ( n,m ) denote A N , thought of as an ( A k ⊗ A ℓ , A n ⊗ A m )-bimodule in the naturalway. Then we have Res A N A k ⊗ A ℓ Ind A N A n ⊗ A m ∼ = ( k,ℓ ) ( A N ) ( n,m ) ⊗ − . On the other hand, for each r satisfying k − m = n − ℓ ≤ r ≤ min { n, k } , we haveInd A k ⊗ A ℓ A r ⊗ A n − r ⊗ A k − r ⊗ A ℓ + r − n Res A n ⊗ A m A r ⊗ A n − r ⊗ A k − r ⊗ A ℓ + r − n ∼ = B r ⊗ − , where B r = ( A k ⊗ A ℓ ) ⊗ A r ⊗ A n − r ⊗ A k − r ⊗ A ℓ + r − n ( A n ⊗ A m ) , and where we view A k ⊗ A ℓ as a right ( A r ⊗ A n − r ⊗ A k − r ⊗ A ℓ + r − n )-module via the map a ⊗ a ⊗ a ⊗ a a a ⊗ a a . (This corresponds to the functor S appearing in (3.6).)Therefore, in order to prove the isomorphism (3.6), it suffices to prove that we have anisomorphism of bimodules ( k,ℓ ) ( A N ) ( n,m ) ∼ = L min { n,k } r = n − ℓ B r . Now, we have one double coset in S k × S ℓ \ S N /S n × S m for each r satisfying k − m = n − ℓ ≤ r ≤ min { n, k } (see, for example, [Zel81, Appendix 3, p. 170]). Precisely, the doublecoset C r corresponding to r consists of the permutations w ∈ S n satisfying | w ( { , . . . , n } ) ∩ { , , . . . , k }| = r, | w ( { n + 1 , . . . , N } ) ∩ { , , . . . , k }| = k − r, | w ( { , . . . , n } ) ∩ { k + 1 , . . . , N }| = n − r, | w ( { n + 1 , . . . , N } ) ∩ { k + 1 , . . . , N }| = ℓ − n + r = m − k + r. Thus the cardinality of the double coset C r is | C r | = m ! n ! (cid:18) kr (cid:19)(cid:18) ℓn − r (cid:19) . The permutation w r ∈ S N given by w r ( i ) = i if 1 ≤ i ≤ r,i − r + k if r < i ≤ n,i − n + r if n < i ≤ n + k − r,i if n + k − r < i ≤ N, is a minimal length representative of C r . It then follows from Lemma 4.2(b) that T w r T i = T i T w r if 1 ≤ i < r or n + k − r < i < N,T w r T i = T i − r + k T w r if r < i < n,T w r T i = T i − n + r T w r if n < i < n + k − r. Thus,(4.2) T w r ( a ⊗ a ⊗ a ⊗ a ) = ( a ⊗ a ⊗ a ⊗ a ) T w r for all a ∈ A r , a ∈ A n − r , a ∈ A k − r , a ∈ A ℓ + r − n .Define B ′ r ⊆ ( k,ℓ ) ( A N ) ( n,m ) to be the sub-bimodule generated by T w r . It follows fromLemma 4.2 that ( k,ℓ ) ( A N ) ( n,m ) = L min { n,k } r = n − ℓ B ′ r and that dim F B ′ r = | C r | . It also follows thatthe dimension of A k ⊗ A ℓ as a right module over A r ⊗ A n − r ⊗ A k − r ⊗ A ℓ − n + r is k ! ℓ ! /r !( n − r )!( k − r )!( ℓ − n + r )! and that the dimension of A m ⊗ A n as a left module over A r ⊗ A n − r ⊗ A k − r ⊗ A ℓ + r − n is m ! n ! /r !( n − r )!( k − r )!( ℓ − n + r )!. Therefore, dim F B r = | C r | = dim F B ′ r .Now consider the ( A k ⊗ A ℓ , A n ⊗ A m )-bimodule map B r → B ′ r uniquely determined by1 A k ⊗ A ℓ ⊗ A n ⊗ A m T w r , which is well defined by (4.2). This map is surjective, and thus is an isomorphism bydimension considerations. (cid:3) Lemma 4.4. If A is a Hecke-like tower of algebras, then we have an isomorphism of functors Ψ ⊗ ∆Ψ − ∼ = S ∆ on A -mod (hence also on A -pmod ).Proof. It suffices to prove that, for m, n ∈ N , we have an isomorphism of functors(Ψ m ⊗ Ψ n ) ◦ Res A m + n A m ⊗ A n ◦ Ψ − m + n ∼ = S ◦ Res A m + n A n ⊗ A m . ATEGORIFICATION AND HEISENBERG DOUBLES ARISING FROM TOWERS OF ALGEBRAS 19
Describing each functor above as tensoring on the left by the appropriate bimodule, it sufficesto prove that we have an isomorphism of bimodules(4.3) ( A ψm ⊗ A ψn ) ⊗ A m ⊗ A n A ψm + n ∼ = S ⊗ A n ⊗ A m A m + n , where A ψk , k ∈ N , denotes A k , considered as an ( A k , A k )-bimodule with the right actiontwisted by ψ k , and where S is A n ⊗ A m considered as an ( A m ⊗ A n , A n ⊗ A m )-module viathe obvious right multiplication and with left action given by ( a ⊗ a , s ) ( a ⊗ a ) s for s ∈ S , a ∈ A m , a ∈ A n .For k ∈ N , let 1 k denote the identity element of A k and 1 ψk denote this same elementconsidered as an element of A ψk . It is straightforward to show that the map between thebimodules in (4.3) given by(1 ψm ⊗ ψn ) ⊗ ψm + n (1 n ⊗ m ) ⊗ m + n . (and extended by linearity) is a well defined isomorphism. (cid:3) Suppose A is a Hecke-like tower of algebras. If d = 0 in Definition 4.1, then T i ( T i − c ) /d =( T i − c ) T i /d = 1 and so T i is invertible for all i . It follows that T w is invertible for all w ∈ S n .On the other hand, if d = 0, then T i ( T i − c ) = 0 and so T i is a zero divisor, hence notinvertible. Therefore, d = 0 if and only if T w is invertible for all w . Lemma 4.5. If A is a Hecke-like tower of algebras with d = 0 (equivalently, such that T i is invertible for all i ), then we have isomorphisms of functors ∇ ∼ = ∇ S and ∆ ∼ = S ∆ on A -mod (hence also on A -pmod ). In particular, A is dualizing.Proof. Let m, n ∈ N and define w ∈ S m + n by w ( i ) = ( m + i if 1 ≤ i ≤ n,i − n if n < i ≤ m + n. Then T w is invertible and, by Lemma 4.2(b), we have T w T i = T w ( i ) T w for all i = 1 , . . . , m − , m + 1 , . . . , m + n −
1. Now, let S be A m ⊗ A n considered as an ( A n ⊗ A m , A m ⊗ A n )-modulevia the obvious right multiplication and with left action given by ( a ⊗ a , s ) ( a ⊗ a ) s for s ∈ S , a ∈ A n , a ∈ A m . Thus, we have an isomorphism of functors S ∼ = S ⊗ − . It isstraightforward to verify that the map A m + n → A m + n ⊗ A n ⊗ A m S, a aT w ⊗ (1 ⊗ , is an isomorphism of ( A m + m , A m ⊗ A n )-bimodules. It follows that ∇ ∼ = ∇ S . The proofthat ∆ ∼ = S ∆ is analogous. The final statement of the lemma then follows from Lemma 4.4and Propositions 3.7 and 4.3. (cid:3) Corollary 4.6.
All Hecke-like towers of algebras are strong and dualizing.Proof.
It follows immediately from Lemma 4.4 and Propositions 3.7 and 4.3 that A is strongand that it is dualizing if and only if K ( A ) is cocommutative. By [BLL12, Prop. 6.1], A isisomorphic to either the tower of nilcoxeter algebras, the tower of Hecke algebras at a genericparameter, the tower of Hecke algebras at a root of unity, or the tower of 0-Hecke algebras.(While the statement of [BLL12, Prop. 6.1] is over C , the proof is valid over an arbitraryalgebraically closed field.) For the tower of Hecke algebras at a generic parameter or a root of unity, we have d = 0 in Definition 4.1 and so K ( A ) is cocommutative by Lemma 4.4. Forthe other two towers, we will see in Sections 5 and 8 that K ( A ) is cocommutative. (cid:3) Remark 4.7.
In this section, the assumption that F is algebraically closed was only usedto conclude that axiom (TA4) of Definition 3.1 is satisfied and in the proof of Corollary 4.6.If F is not algebraically closed but the bilinear form (3.4) is still a perfect pairing, then allthe results of this section remain true except that one must replace Corollary 4.6 by thestatement that the tower of nilcoxeter algebras, the tower of Hecke algebras at a genericparameter, the tower of Hecke algebras at a root of unity, and the tower of 0-Hecke algebrasare all strong and dualizing. 5. Nilcoxeter algebras
In this section we specialize the constructions of Sections 2 and 3 to the tower of nilcoxeteralgebras of type A . We will see that we recover Khovanov’s categorification of the polynomialrepresentation of the Weyl algebra (see [Kho01]). We let F be an arbitrary field.The nilcoxeter algebra N n is the Hecke-like algebra of Definition 4.1 with c = d = 0. Therepresentation theory of N n is straightforward. (We refer the reader to [Kho01] for proofsof the facts stated here.) Up to isomorphism, there is one simple module L n , which is onedimensional, and on which the generators all act by zero. The projective cover of L n is P n = N n , considered as an N n -module by left multiplication. We have isomorphisms of Hopfalgebras K ( N ) ∼ = Z [ x ] , [ P n ] x n , G ( N ) ∼ = Z [ x, x / , x / , . . . ] ⊆ Q [ x ] , [ L n ] x n /n ! . In both cases, the coproduct is given by ∆( x ) = x ⊗ ⊗ x . We also have G proj ( N ) ∼ = Z [ x ] , and the Cartan map K ( N ) → G ( N ) of Definition 3.9 corresponds to the natural inclusion Z [ x ] ֒ → Z [ x, x / , x / , . . . ].The inner product satisfies h x m , x n n ! i = h [ P m ] , [ L n ] i = δ mn . Therefore x ∗ (cid:0) x m m ! (cid:1) = x m − ( m − , i.e. x ∗ = ∂ x corresponds to partial derivation by x . Therefore the algebra h in this setting is thesubalgebra of End Z [ x, x / , x / , . . . ] generated by x, x / , x / , . . . and ∂ x . In addition, h proj is the algebra generated by x and ∂ x , with relation [ ∂ x , x ] = 1. The Fock space F proj is the representation of h proj given by its natural action on Z [ x ]. It follows from the abovethat Q ⊗ Z h ∼ = Q ⊗ Z h proj is the rank one Weyl algebra.By Remark 4.7, the tower N is strong and dualizing. Thus, Theorem 3.18 providesa categorification of the polynomial representation of the Weyl algebra. In fact, (3.14)specializes to the main result of [Kho01] if one takes M and P to be the trivial N -modules.Indeed, with these choices we have Ψ ⊗ ∆Ψ − ( P ) = ( F ⊗ F ) ⊕ ( F ⊗ F ), where F i denotesthe trivial N i -module for i = 0 ,
1. Then (3.14) becomesRes N n +1 N n ◦ Ind N n +1 N n ∼ = (cid:16) Ind N n N n − ◦ Res ( F ⊗ F ) ( F ⊗ − ) (cid:17) ⊕ (cid:0) Res ( F ⊗ F ) ( M ⊗ − ) (cid:1) ∼ = (cid:16) Ind N n N n − ◦ Res N n N n − (cid:17) ⊕ Id , ATEGORIFICATION AND HEISENBERG DOUBLES ARISING FROM TOWERS OF ALGEBRAS 21 which is the categorification of the relation ∂ x x = x∂ x + 1 appearing in [Kho01, (13)]. (Notethat while [Kho01] works over the field Q , the arguments go through over more general F .)6. Hecke algebras at generic parameters
In this section we specialize the constructions of Sections 2 and 3 to the tower of algebrascorresponding to the Hecke algebras of type A at a generic parameter. The results of thissection also apply to the group algebra of the symmetric group (the case when q = 1).6.1. The Hecke algebra and symmetric functions.
Let A n be the Hecke algebra at ageneric value of q . More precisely, assume q ∈ C × is not a nontrivial root of unity and let A n be the unital C -algebra of Definition 4.1 with c = q − d = q . By convention, we set A = A = C . Then A = L n ∈ N A n is a Hecke-like tower of algebras. It is well known that acomplete set of irreducible A n -modules is given by { S λ | λ ∈ P ( n ) } , where S λ is the Spechtmodule corresponding to the partition λ (see [DJ86, § A n are semisimple, wehave K ( A ) = G ( A ). In fact, both are isomorphic (as Hopf algebras) to Sym, the algebraof symmetric functions in countably many variables x , x , . . . over Z . This isomorphism isgiven by the map sending [ S λ ] to s λ , the Schur function corresponding to the partition λ (see, for example, [Zel81]). Recall that Sym is a graded connected Hopf algebra:Sym = L n ≥ Sym n , where Sym n is the Z -submodule of Sym consisting of homogeneous polynomials of degree n .We adopt the convention that Sym n = 0 for n <
0. The inner product (3.4) corresponds tothe usual inner product on Sym under which the Schur functions are self-dual. Furthermore,the monomial and homogeneous symmetric functions are dual to each other: h m λ , h µ i = δ λ,µ , λ, µ ∈ P . Under this inner product, Sym is self-dual as a Hopf algebra. In other words, (Sym , Sym) isa dual pair of Hopf algebras.6.2.
The Heisenberg algebra.
Applying the construction of Section 2.2 to the dual pair(Sym , Sym), we obtain the
Heisenberg algebra h = h (Sym , Sym). We obtain a minimalpresentation of h by considering two collections of polynomial generators for Sym (one forSym viewed as H + and one for Sym viewed as H − ). In particular, if we choose the powersum symmetric functions p n , n ∈ N + , in both cases, we get the usual presentation of theHeisenberg algebra:[ p n , p k ] = 0 , [ p ∗ n , p ∗ k ] = 0 , [ p ∗ n , p k ] = nδ n,k , n, k ∈ N + . However, { p ∗ n , p n | n ∈ N + } is only a generating set for h ⊗ Z Q since the power sum symmetricfunctions only generate the ring of symmetric functions over Q .One the other hand, if we choose the elementary symmetric functions e n , n ∈ N + , andthe complete symmetric functions h n , n ∈ N + , we have the following relations:(6.1) [ e n , e k ] = 0 , [ h ∗ n , h ∗ k ] = 0 , [ h ∗ n , e k ] = e k − h ∗ n − , n, k ∈ N + . (We adopt the convention that e = e ∗ = h = h ∗ = 1 and e n = e ∗ n = h n = h ∗ n = 0 for n < h (one does not need to tensor with Q ) and is the oneused in the categorification of h given in [Kho, LS13] (for an overview, see [LS12]). Other choices of polynomial generators result in different presentations. For the sake ofcompleteness we record the other nontrivial relations:(6.2) [ e ∗ n , h k ] = h k − e ∗ n − , [ h ∗ n , h k ] = P i ≥ h k − i h ∗ n − i , [ e ∗ n , e k ] = P i ≥ e k − i e ∗ n − i , n, k ∈ N + . To prove these relations, we use the fact (see, for example, [Zab, Prop. 3.6]) that, for k, n ∈ N ,we have h ∗ k ( h n ) = h n − k , h ∗ k ( e n ) = ( δ k + δ k ) e n − k , h ∗ k ( p n ) = δ kn + δ k p n ,e ∗ k ( h n ) = ( δ k + δ k ) h n − k , e ∗ k ( e n ) = e n − k , e ∗ k ( p n ) = ( − k − δ kn + δ k p n ,p ∗ k ( h n ) = h n − k , p ∗ k ( e n ) = ( − k − e n − k , p ∗ k ( p n ) = nδ nk + δ k p n . Then, for example, since ∆( e n ) = P ni =0 e i ⊗ e n − i , we have, by Lemma 2.8, e ∗ n ( h k f ) = P ni =0 e ∗ i ( h k ) e ∗ n − i ( f ) = h k e ∗ n ( f ) + h k − e ∗ n − ( f ) for all f ∈ Sym . Thus [ e ∗ n , h k ] = h k − e ∗ n − . The other relations are proven similarly.6.3. Categorification.
By Corollary 4.6, the tower A is strong and dualizing. For n ∈ N ,let E n (resp. L n ) be the one-dimensional representation of A n on which each T i acts by − q ). Then ∆( E n ) ∼ = P ni =0 E i ⊗ E n − i and ∆( L n ) ∼ = P ni =0 L i ⊗ L n − i . SinceHom A n ( L n , E n ) = 0 unless n = 0 or n = 1 (in which case E n and L n are both the trivialmodule), we have, by (3.14),Res L n ◦ Ind E k ∼ = ∇ (cid:0)L ni =0 Res L i ⊗ L n − i ( E k ⊗ − ) (cid:1) ∼ = (Ind E k ◦ Res L n ) ⊕ (cid:0) Ind E k − ◦ Res L n − (cid:1) , which is a categorification of the last relation of (6.1) since, under the isomorphism G ( A ) ∼ = K ( A ) ∼ = Sym, the class of the representation L n corresponds to h n and the class of E k corresponds to e k . By (3.12), (3.13), and Lemma 4.5, we haveInd E n ◦ Ind E k ∼ = Ind ∇ ( E n ⊗ E k ) ∼ = Ind ∇ ( E k ⊗ E n ) ∼ = Ind E k ◦ Ind E n , andRes L n ◦ Res L k ∼ = Res ∇ ( L n ⊗ L k ) ∼ = Res ∇ ( L k ⊗ L n ) ∼ = Res L k ◦ Res L n , which categorifies the first two relations of (6.1). Remark 6.1.
While we have chosen to show how Theorem 3.18 recovers a categorificationof the relations (6.1), we could just have easily used it to recover categorifications of therelations (6.2). This is an illustration of the fact that Theorem 3.18 does not rely on aparticular presentation of the Heisenberg double h ( A ). Remark 6.2.
The special case of Theorem 2.11(c) for the dual pair (Sym , Sym) is knownas the
Stone–von Neumann Theorem . Remark 6.3.
Since we have K ( A ) = G ( A ), it follows that h proj = h in this case (seeDefinition 3.13). 7. Hecke algebras at roots of unity
We now consider Hecke algebras at a root of unity. Fix ℓ ∈ N + and consider the unital C -algebra A n with generators and relations as in Section 6.1, but with q replaced by a fixed ℓ th root of unity ζ . By Corollary 4.6, A = L n ∈ N A n is a strong dualizing tower of algebras.We refer the reader to [LLT96, § ATEGORIFICATION AND HEISENBERG DOUBLES ARISING FROM TOWERS OF ALGEBRAS 23
Let J ℓ ⊆ Sym be the ideal generated by the power sum symmetric functions p ℓ , p ℓ , p ℓ , . . . ,and let J ⊥ ℓ be its orthogonal complement relative to the standard inner product on Sym(see Section 6.1). Then there are isomorphisms of Hopf algebras K ( A ) ∼ = J ⊥ ℓ and G ( A ) ∼ = Sym / J ℓ . Moreover, under these identifications, the inner product between K ( A ) and G ( A ) is thatinduced by the standard inner product on Sym.Recall that a partition λ is said to be ℓ -regular if each part appears fewer than ℓ times.The specialization ¯ S λ of the Specht module S λ , λ ∈ P ( n ), to q = ζ is, in general, no longeran irreducible A n -module. However, it was shown in [DJ86, §
6] that if λ is ℓ -regular, then¯ S λ contains a unique maximal submodule rad ¯ S λ . As λ varies over the ℓ -regular partitions of n , D λ := ¯ S λ / rad ¯ S λ varies over a complete set of nonisomorphic irreducible representationsof A n . It follows that a basis of G ( A ) (resp. K ( A )) is given by the [ D λ ] (resp. [ P λ ], where P λ is the projective cover of D λ ) as λ varies over the set of ℓ -regular partitions. In theory, onecould compute the relations in h ( A ) in these bases by using the results of [LLT96] to expressthe basis elements in terms of the standard symmetric functions and then use the relationsin Section 6.2. In this way, one would obtain a presentation of h ( A ). Of course, in general,this presentation would be far from minimal.In fact, it turns out that h ( A ) is an integral from of the usual Heisenberg algebra h (Sym , Sym) (see Section 6.2). This can be seen as follows. Recall that the set of powersum functions p λ , λ ∈ P , is an orthogonal basis of Sym Q = Q ⊗ Z Sym. (Throughout we usea subscript Q to denote extension of scalars to the rational numbers.) Therefore J ℓ, Q has abasis given by the set { p λ | ℓ divides λ i for at least one i } , and J ⊥ ℓ, Q has a basis { p λ | ℓ does not divide λ i for any i } . Similarly, (Sym / J ℓ ) Q has a basis { p λ + J ℓ | ℓ does not divide λ i for any i } . Remark 7.1.
We see from the above that G ( A n ) and K ( A n ) have bases indexed, on the onehand, by the set of ℓ -regular partitions of n and, on the other hand, by the set of partitionsof n in which no part is divisible by ℓ . A correspondence between these two sets of partitionsis given by Glaisher’s Theorem (see, for example, [Leh46, p. 538]).For m ∈ N + such that ℓ does not divide m , let q m = p m + J ℓ . Then we have algebraisomorphisms J ⊥ ℓ, Q ∼ = Q [ p m | m ∈ N + , ℓ does not divide m ] , and(Sym / J ℓ ) Q ∼ = Q [ q m | m ∈ N + , ℓ does not divide m ] . Thus, h ( A ) Q is generated by { p m , q m | m ∈ N + , ℓ does not divide m } subject to the relations[ p m , p n ] = [ q m , q n ] = 0 and [ p m , q n ] = mδ m,n
1. It follows that h ( A ) Q is isomorphic asan algebra to the classical Heisenberg algebra h (Sym , Sym) Q . Thus we have the followingproposition. Proposition 7.2.
The Heisenberg double associated to the tower of Hecke algebras at a rootof unity is an integral form of the classical Heisenberg algebra: h ( A ) Q ∼ = h (Sym , Sym) Q . By Proposition 7.2, as ℓ varies over the positive integers, we obtain a family of integralforms of the classical Heisenberg algebra. It would be interesting to work out minimalpresentations of these integral forms over Z . Furthermore, the Cartan map K ( A ) → G ( A )is known to have a nonzero determinant (see [BK02, Cor. 1]). Therefore, it induces anisomorphism G proj ( A ) Q ∼ = G ( A ) Q , which implies that h proj ( A ) Q ∼ = h ( A ) Q . It is not knownwhether h proj ( A ) ∼ = h ( A ).It is known that the category A -pmod yields a categorification of the basic representationof b sl n via i -induction and i -restriction functors (see [LLT96, p. 218]). Theorem 3.18 providesa categorification of the principle Heisenberg subalgebra of b sl n .8. We now specialize the constructions of Sections 2 and 3 to the tower of 0-Hecke algebrasof type A . We begin by recalling some basic facts about the rings of quasisymmetric andnoncommutative symmetric functions. We refer the reader to [LMvW] for further details.8.1. The quasisymmetric functions.
Let QSym be the algebra of quasisymmetric func-tions in the variables x , x , . . . over Z . Recall that this is the subalgebra of Z J x , x , . . . K consisting of shift invariant elements. That is, f ∈ QSym if and only if, for all k ∈ N + , thecoefficient in f of the monomial x n x n · · · x n k k is equal to the coefficient of the monomial x n i x n i · · · x n k i k for all strictly increasing sequences of positive integers i < i < · · · < i k andall n , n , . . . , n k ∈ N . The algebra QSym is a graded algebra:QSym = L n ≥ QSym n , where QSym n is the Z -submodule of QSym consisting of homogeneous elements of degree n .We adopt the convention that QSym n = 0 for n < monomial quasisymmetric functions M α ,which are indexed by compositions α = ( α , . . . , α r ) ∈ C : M α = P i < ···
1) and β = (3 , α · β =(1 , , , ,
5) and α ⊙ β = (1 , , , M α ) = P α = β · γ M β ⊗ M γ , ATEGORIFICATION AND HEISENBERG DOUBLES ARISING FROM TOWERS OF ALGEBRAS 25 ∆( F α ) = P α = β · γ or α = β ⊙ γ F β ⊗ F γ . Note that naturally Sym ⊆ QSym. In particular, the monomial symmetric functions canbe handily expressed in terms of the monomial quasisymmetric functions:(8.1) m λ = P e α = λ M α , where e α is the partition obtained by sorting α. The noncommutative symmetric functions.
Define NSym, the algebra of noncom-mutative symmetric functions , to be the free associative algebra (over Z ) generated by thealphabet h , h , . . . . Thus NSym has a basis given by h α := h α · · · h α r , α ∈ C . This is agraded algebra: NSym = L n ≥ NSym n , where NSym n = Span { h α | α ∈ C ( n ) } . We adopt the convention that NSym n = 0 for n < noncommutative ribbon Schur functions r α are defined to be r α = P α (cid:22) β ( − ℓ ( α ) − ℓ ( β ) h β , α ∈ C . These basis elements multiply nicely: h α h β = h α · β and r α r β = r α · β + r α ⊙ β . In fact, NSym is a graded connected Hopf algebra. The coproduct is given by the formula(8.2) ∆( h n ) = P ni =0 h i ⊗ h n − i . The 0-Hecke algebra and its Grothendieck groups.
Let F be an arbitrary fieldand let A n be the unital F -algebra with generators and relations as in Section 6.1, but with q replaced by 0 (i.e. the 0-Hecke algebra). Consider the tower of algebras A = L n ∈ N A n .The irreducible A n -modules are all one-dimensional and are naturally enumerated by theset C ( n ) of compositions of n (see [Nor79, §
3] and [KT97, § L α be the irreduciblemodule corresponding to the composition α ∈ C ( n ) and let P α be its projective cover. Wethen have (see [KT97, Cor. 5.8 and Cor. 5.11] – while the statements there are for the casethat F = C , the proofs remain valid over more general fields) H − = K ( A ) ∼ = NSym , [ P α ] r α , (8.3) H + = G ( A ) ∼ = QSym , [ L α ] F α . (8.4)We also have G proj ( A ) ∼ = Sym , and the Cartan map K ( A ) → G proj ( A ) of Definition 3.9 corresponds to the projection of Hopfalgebras(8.5) χ : NSym ։ Sym , h α h e α . Alternatively, it is given by χ ( r α ) = r α , where r α is the usual ribbon Schur function. This isa reformulation of [KT97, Prop. 5.9].The bilinear form (3.4) becomes the well-known perfect Hopf pairing of the Hopf algebrasQSym and NSym given as follows: h· , ·i : NSym × QSym → Z , h h α , M β i = δ αβ = h r α , F β i , α, β ∈ C . In this way, (NSym , QSym) is a dual pair of Hopf algebras.
The quasi-Heisenberg algebra.
We now apply the construction of Section 2.2 to thedual pair (QSym , NSym).
Definition 8.1 ((Projective) quasi-Heisenberg algebra) . We call q := h (QSym , NSym) the quasi-Heisenberg algebra . We define the projective quasi-Heisenberg algebra q proj to be thesubalgebra of q generated by NSym and Sym ⊆ QSym (see Definition 3.13).
Lemma 8.2. In q we have, for all α = ( α , . . . , α r ) ∈ C , n ∈ N + , (cid:2) R h ∗ n , M α (cid:3) = M ( α ,...,α r − ) R h ∗ n − α r , with the understanding that R h ∗ k = 0 for k < .Proof. By Lemma 2.8 and (8.2), we have R h ∗ n ( M α G ) = P ni =0 R h ∗ i ( M α ) R h ∗ n − i ( G ) . So, if n ≥ α r , we have R h ∗ n ( M α G ) = M α R h ∗ n ( G ) + M ( α ,...,α r − ) R h ∗ n − α r ( G ). The resultfollows. (cid:3) Corollary 8.3.
For n ∈ N and λ ∈ P , we have (8.6) [ R h ∗ n , m λ ] = P nj =1 m λ − j R h ∗ n − j , where λ − j is equal to the partition obtained from removing a part j from λ if λ has such apart and m λ − j is defined to be zero otherwise. In particular, for n, k ∈ N , we have [ R h ∗ n , p k ] = R h ∗ n − k , [ R h ∗ n , e k ] = e k − R h ∗ n − , [ R h ∗ n , h k ] = P nj =1 h k − j R h ∗ n − j . Proof.
Equation (8.6) follows from (8.1) and Lemma 8.2. The remainder of the relations thenfollow by expressing p k , e k and h k in terms of the monomial symmetric functions m λ . (cid:3) Corollary 8.4.
The quasi-Heisenberg algebra q is generated by the set { M α , R h ∗ n | α ∈ C , n ∈ N + } . The M α multiply as in QSym (for a precise description of this product, see [LMvW, § )and (cid:2) R h ∗ n , M α (cid:3) = M ( α ,...,α r − ) R h ∗ n − α r , α = ( α , . . . , α r ) ∈ C , n ∈ N + . Remark 8.5.
We also note that the algebra q has generators { F α , R h ∗ n | α ∈ C , n ∈ N + } . From a representation theoretic point of view, these are more natural since the F α correspondto simple A n -modules (see (8.4)). The F α then multiply as in QSym (for a precise descriptionof this product, see [LMvW, § (cid:2) R h ∗ n , F α (cid:3) = P α r i =1 F ( α ,...,α r − i ) R h ∗ n − i , α = ( α , . . . , α r ) ∈ C , n ∈ N + . Remark 8.6.
Note that the above presentations are far from minimal. There are polynomialgenerators of QSym, enumerated by elementary Lyndon words (see [HGK10, Th. 6.7.5]),which one could use instead of the M α in the above presentation. This would result in aminimal presentation of q .The following result gives a presentation of the projective quasi-Heisenberg algebra interms of generators and relations. ATEGORIFICATION AND HEISENBERG DOUBLES ARISING FROM TOWERS OF ALGEBRAS 27
Proposition 8.7.
The algebra q proj is generated by the set { e n , R h ∗ n | n ∈ N } . The relations are [ e n , e k ] = 0 , [ R h ∗ n , e k ] = e k − R h ∗ n − , n, k ∈ N . Proof.
This follows immediately from the definition of q proj and Corollary 8.3. (cid:3) Remark 8.8.
Note the similarity of the presentation of Proposition 8.7 to the presentationof the usual Heisenberg algebra h (Sym , Sym) given in (6.1). The only difference is thatthe h ∗ n commute, whereas the R h ∗ n do not. There is a natural surjective map of algebras q proj → h (Sym , Sym) given by e n e n , R h ∗ n h ∗ n , n ∈ N + .8.5. Fock spaces and categorification.
As described in Section 2.3, the quasi-Heisenbergalgebra q acts naturally on QSym and we call this the lowest weight Fock space representation of q . By Theorem 2.11(c), any representation of q generated by a lowest weight vacuum vectoris isomorphic to QSym.Similarly, as in Definition 3.14, the projective quasi-Heisenberg algebra q proj acts natu-rally on Sym and we call this the lowest weight Fock space representation of q proj . As a q proj -module, Sym is generated by the lowest weight vacuum vector 1 ∈ Sym. By Propo-sition 3.15(c), any representation of q proj generated by a lowest weight vacuum vector isisomorphic to Sym. However, this representation is not faithful since it factors through theprojection from q proj to the usual Heisenberg algebra (see Remark 8.8). On the other hand,the highest weight Fock space representation of q proj is faithful (see Proposition 3.16).By Remark 4.7, A is a strong dualizing tower of algebras. Therefore, Theorem 3.18yields a categorification of the Fock space representations of q and q proj . For instance, it isstraightforward to verify that∆( L α ) ∼ = L α = β · γ or α = β ⊙ γ L β ⊗ L γ for all α ∈ C , Ψ ⊗ ∆Ψ − ( P ( n ) ) ∼ = ∆( P ( n ) ) ∼ = L ni =0 P ( i ) ⊗ P ( n − i ) for all n ∈ N . For α = ( α , . . . , α r ) ∈ C and i ∈ { , . . . , α r } , it follows that Res P ( i ) L α = L ( α ,...,α r − i ) . Thuswe haveRes P ( n ) ◦ Ind L α ∼ = ∇ (cid:16)L ni =0 Res P ( i ) ⊗ P ( n − i ) ( L α ⊗ − ) (cid:17) ∼ = L α r i =0 Ind L ( α ,...,αr − i ) Res P ( n − i ) , which is a categorification of the relation (8.7). The categorification of the multiplication ofthe elements F α , α ∈ C , follows from the computation of the induction in A -mod (see, forexample, the proof of [DKKT97, Prop. 4.15]).9. Application:
QSym is free over
SymAs a final application of the methods of the current paper, we use the generalized Stone–von Neumann Theorem for q proj (Proposition 3.15) to prove that QSym is free over Sym.This gives a proof that is quite different from the one previously appearing in the literature.The previous proof proceeds by constructing a free commutative polynomial basis of QSymenumerated by elementary Lyndon words (see [HGK10, Thm 6.7.5]). Then the freeness ofQSym over Sym follows from the fact that the polynomial basis contains the elementarysymmetric functions. The construction of this polynomial basis turned out to be rather difficult, and many false proofs appeared in the literature. We refer the reader to [Haz01a,Haz01b] for more on the history of this result. Lemma 9.1.
Suppose V is a q proj -module which is generated (as a q proj -module) by a finiteset of lowest weight vacuum vectors. Then V is a direct sum of copies of lowest weight Fockspace.Proof. Let { v i } i ∈ I denote a set of lowest weight vacuum vectors that generates V and suchthat I has minimal cardinality. We claim that(9.1) Z v i ∩ Z v j = { } for all i = j. Suppose, on the contrary, that Z v i ∩ Z v j = { } for some i = j . Then n i v i = n j v j for some n i , n j ∈ Z . Let m = gcd( n i , n j ) and choose a i , a j ∈ Z such that m = a i n i + a j n j . Set w = a j v i + a i v j . Then w is clearly a lowest weight vacuum vector, and we have n i m w = 1 m ( a j n i v i + a i n i v j ) = 1 m ( a j n j v j + a i n i v j ) = v j . Similarly, n j m w = v i . Thus { v k } k ∈ I \{ i,j } ∪ { w } is a set of lowest weight vacuum vectors thatgenerates V , contradicting the minimality of the cardinality of I .By Proposition 3.15(c), q proj · v i ∼ = Sym as q proj -modules. It then follows from Proposi-tion 3.15(a) and (9.1) that q proj · v i ∩ q proj · v j = { } for i = j . The lemma follows. (cid:3) Define an increasing filtration of q proj -submodules of QSym as follows. For n ∈ N , letQSym ( n ) := P ℓ ( α ) ≤ n q proj · M α . In particular, note that QSym (0) = Sym. We adopt the convention that QSym ( − = { } . Proposition 9.2.
The space
QSym of quasisymmetric functions is free as a
Sym -module.Proof.
Note that, for α ∈ C such that ℓ ( α ) = n , we have R h ∗ m ( M α ) ∈ QSym ( n − for any m >
0. Therefore, in the quotient V n = QSym ( n ) / QSym ( n − , such M α are lowest weightvacuum vectors. It is clear that these vectors generate V n , and therefore, by Lemma 9.1, V n = L v ∈S n Sym · v, where S n is some collection of vacuum vectors in V n .Consider the short exact sequence0 → QSym ( n − → QSym ( n ) → V n → . Since V n is a free (hence projective) Sym-module, the above sequence splits. Therefore, ifQSym ( n − is free over Sym, then so is QSym ( n ) .By the argument in the previous paragraph we can choose nested sets of vectors in QSym e S ⊆ e S ⊆ e S ⊆ · · · such that, for every n ∈ N , we have QSym ( n ) = L ˜ v ∈ e S n Sym · ˜ v . Let e S = S n ∈ N e S n . ThenQSym = L v ∈ e S Sym · v. (cid:3) ATEGORIFICATION AND HEISENBERG DOUBLES ARISING FROM TOWERS OF ALGEBRAS 29
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A. Savage: Department of Mathematics and Statistics, University of Ottawa, Canada
URL : http://mysite.science.uottawa.ca/asavag2/ E-mail address : [email protected] O. Yacobi: School of Mathematics and Statistics, University of Sydney, Australia
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