Quantum polynomial functors from $e$-Hecke pairs

We define a new category of quantum polynomial functors extending the quantum polynomials introduced by Hong and Yacobi. We show that our category has many properties of the category of Hong and Yacobi and is the natural setting in which one can define composition of quantum polynomial functors. Throughout the paper we highlight several key differences between the theory of classical and quantum polynomial functors.


Introduction
Hong and Yacobi [10] introduced a category of quantum polynomial functors which quantizes the strict polynomial functors of Friedlander and Suslin [7]. The purpose of this paper is to introduce higher level categories of quantum polynomial functors that extend the construction of Hong and Yacobi and explain why they give a natural quantization of classical polynomial functors. The most visible advantage of our definition is that we are now able to compose quantum polynomial functors.
A polynomial functor is defined as a functor between vector spaces which is polynomial on the space of morphisms. The definition can be formulated as follows. Let k be a field, and let V be the category of finite dimensional vector spaces over k. The dth symmetric group S d acts on V ⊗d by permuting tensor factors. Let d V be the category which has the same objects as V does while the set of morphisms between V , W ∈ V is The category P d of polynomial functors of degree d is the category of linear functors F : d V → V. A polynomial functor is by definition an object in the category P = d≥0 P d . The category P has initially been introduced by [7] to prove the cohomological finite generation of finite group schemes over a field, where they use certain rational cohomology computations for the general linear groups effectively done in the category of polynomial functors. Another example of important rational cohomology result obtained from the polynomial functors is the untwisting of Frobenius due to Chałupnik [2] and Touzé [16]. Polynomial functors can also be used to compute cohomology for other classical algebraic groups (see [15] and references therein). In a different direction, Hong, Touzé and Yacobi [8,9] showed that there is a categorical action of sl p on P that categorifies the action of sl p on the Fock space, where p is the characteristic of the base field.
Two instrumental properties in the theory of polynomial functors are representability and composition. Representability allows one to prove that the category P d is equivalent to the category of modules of the Schur algebra S(n; d) for any n ≥ d, or the category of polynomial GL n representations of degree d ( [7,Sect. 3]). Composition is a natural property of functors, which is not so natural for modules over a Schur algebra, hence is a main advantage of studying polynomial functors. It is extremely useful in performing cohomology calculations. Given F, G ∈ P and an exact sequence in P that represents an element of Ext n P (F, G), precomposing (respectively, postcomposing, when it is exact) the exact sequence with another polynomial functor H ∈ P gives a class in Ext n P (F H, G H) (resp., Ext n P (H F, H G)). The special case of precomposing by the Frobenius functor, where the assignment is injective, is particularly interesting in many contexts (see [5]). In general, this can relate Ext spaces in different degrees and provides, for example [7,Theorem 2.13] which is a key technique in main computations in [7].
A natural question is whether one can deform the polynomial functors into quantum polynomial functors. A first such quantization is due to Hong and Yacobi [10]. They introduced a new category P d q that is a q-deformation of the category P d and showed that it enjoys many properties that P d has. As in the classical case where the corresponding category is equivalent to the module category for the Schur algebra, P d q is equivalent to the category of finite dimensional modules for the quantum Schur algebra S q (n; d) of Dipper and James [4]. They present several applications to quantum invariant theory, including a quantum version of (GL n , GL m )-Howe duality. However, their category of quantum polynomial functors does not allow for composition of the functors.
The underlying reason is that the action of S d in Eq. (1) is replaced by a quantum action that depends on the extra structure given to the objects. In [10], the domain category of P d q consists of pairs (V n , R n ), where one can think of V n as the defining n-dimensional U q (gl n )module or the defining A q (n, n)-comodule (where A q (n, n) is the quantum coordinate ring of n ×n matrices). The generators of the braid group B d act on V ⊗d n via the standard R-matrix R n , the R-matrix of the defining U q (gl n )-representation V n . When one applies a quantum polynomial F to V n , there is an U q (gl n ) structure on F(V n ) which produces an R-matrix R F(V n ) associated to F(V n ). The problem is then that R F(V n ) is not a standard R-matrix, so the pair (F(V ), R F(V ) ) is not in the domain category.
We can also see this in the classical case. The domain of a polynomial functor consists of vector spaces. One can endow such vector spaces with GL n -module structures. If V is a polynomial representation of degree e, then F ∈ P d q maps V to a polynomial representation F(V ) of degree de. So we can think of the domain in the classical case as containing finite dimensional polynomial representations of GL n . The reason why this extra structure of V is not present in the definition of polynomial functors is because S d acts on V ⊗d in the same way regardless of the GL n -structure on V . In other words, one can view a single polynomial functor as a functor between degree e modules and degree de modules for all e ∈ N at the same time. This is not true for quantum polynomial functors. The braid group acts on V ⊗d via the R-matrix of V , and different U q (gl n )-modules have different associated R-matrices.
Therefore, the action of B d on V ⊗d depends on the module structure of V . If V is the defining representation, the action of the braid group on V ⊗d factors through the Hecke algebra H d , but if V is the eth tensor power of the defining representation, or its subquotient" this action factors through the e-Hecke algebra H d,e (see Definition 2.9). The e-Hecke algebras are quantizations of the symmetric group and together play the role which the symmetric group plays in the classical case.
This extra structure should be taken into account in the quantum case. It is then natural to introduce the categories P d q,e of quantum polynomial functors that map degree e modules to degree de modules. In this category, we are able to compose quantum polynomials by Theorem 5.2. More precisely, given G ∈ P d 2 q,e and F ∈ P d 1 q,d 2 e , their composition F • G is a polynomial in P d 1 d 2 q,e . We think of the categories P d q,e as "higher" analogues of P d q . They satisfy many of the properties that P d q satisfy. For example we show in Theorem 4.4 that the category P q,e = d P d q,e is a braided monoidal category. Another fundamental property we prove is that P d q,e has a (finite) generator when q is generic (Theorem 6.13). The root of unity case is significantly harder than the generic q case, since the domain category of the functors in P d q,e consists of degree e polynomial representations of U q (gl n ), which is more complicated then the corresponding category for generic q. However, when e = 1, the category of polynomial representations of degree 1 is "the same" for q a root of unity and for generic q; it consists only of direct sums of the defining representation. Therefore our finite generation result holds for any q when e = 1.
The generator in Theorem 6.13 is defined in terms of a direct sum even for e = 1 (when we would hope our category to reduce to the category studied by Hong and Yacobi). This is something that is needed in order to define composition in full generality as we explain in Sect. 6. We then consider another category of polynomial functors, whose definition involves restricting the domain; we denote the new category by P •,d q,e . This category has a projective generator as shown in Theorem 6.15 and we explain that when e = 1 we get back the main result of [10] in full generality in Remark 6.17.
The existence of the projective generators allows us to conclude the equivalence between the categories P •,d q,e and P d q,e and the category of finite dimensional modules of certain Schur algebra. Such Schur algebras are natural generalizations of the quantum Schur algebra S q (n; d); we conjecture they also appear in a generalization of quantum Schur-Weyl duality.
Another interesting difference between the quantum and the classical categories can be seen when taking into consideration e-Hecke pairs for all e at once. In Sect. 7, we define the category P d q as the category of functors with domain all e-Hecke pairs for all e. For q = 1, this category is equivalent to the category P d q,e for any e. When q is generic, P d q is not equivalent to P d q,e for any e. To further endorse the categories P d q,e as quantum analogues of strict polynomial functors we give several examples of objects in P d q,e . The most interesting we believe are the quantum symmetric and exterior powers of Berenstein and Zwicknagl [1].
We now briefly outline the structure of the paper. In Sect. 2, we introduce the basics of quantum multilinear algebra which are of use throughout the paper. In Sect. 3, we define the categories P d q,e , the main objects to be studied in this paper and present several interesting examples of quantum polynomial functors. We focus on the quantum divided, symmetric and exterior power (due to Berenstein and Zwicknagl [1]). We show that the definition of these objects produce quantum polynomial functors, so their construction fits in our framework. In Sect. 4, we show that the category of quantum polynomial functors is a braided monoidal category. In Sect. 5, we explain how two quantum polynomial functors can be composed in our setting. In Sect. 6, we show that the categories P d q,e and P •,d q,e have a (finite) projective generator for generic q. This immediately implies equivalence to the category of modules of a "generalized" q-Schur algebra. In Sect. 7, we consider a different category P d q with domain all e-Hecke pairs for all e ≥ 0 and we show that this category does not contain a projective generator for generic q. In Sect. 8, we discuss quantum polynomial functors at roots of unity.

Preliminaries
Let k be a field. Let q be an element of k × . We say q is generic if q is not a root of unity. In this section we introduce several objects and prove some properties which will be of use in defining quantum polynomial functors. We note that several of these definitions and some of the properties are taken directly from [10].

Yang-Baxter spaces
Let V be the category of finite dimensional vector spaces over the field k. Each V ∈ V comes with a chosen basis {v 1 , . . . , v n } where n = dim(V ). Even though our results are independent of the chosen basis, the exposition is more clear if we associate a fixed basis to each vector space. Let The Hecke algebra H d is the quotient of the braid group B d by the relations is called an R-matri x if it satisfies the Yang-Baxter equation: where To each pair we can associate a right representation, ρ d, We will most of the time use the short hand notation V for the Yang-Baxter space (V , R) and denote the R-matrix in the pair (V , R) by R := R V .
We now define the quantum Hom-space algebra as it is defined by Hong and Yacobi [10]. Given two Yang-Baxter spaces V and W with basis {v i } and {w j }, respectively, let T (V , W ) be the tensor algebra of Hom(V , W ), that is Denote by x ji : W → V the map

Lemma 2.1
The algebra A(W , V ) has a presentation by the generators x ji and the relations generated by where the coefficients R kl V ,i j are defined by the following equation: Proof Elements of the form x i j x kl form a basis of Hom(W ⊗2 , V ⊗2 ). The quadratic relations (4) are exactly the relations that generate There is a degree preserving morphism of algebras This extends to a map V ,W : V → W ⊗ A(W , V ).
By taking the dual of V ,W ,U we obtain a map There is a natural map m U ,W ,V : The following Proposition shows they are the same map under the isomorphism in Proposition 2.3.

Proposition 2.4
Given three Yang-Baxter spaces V , U , W , the following diagram commutes: Proof See Proposition 2.8 in [10].

Remark 2.5
If W = V , the quadratic relation (4) becomes the RTT relation due to Faddeev, Reshetikhin and Taktajan. The algebra A(V , V ) is then just the algebra denoted by A R V in [6].
We record two properties of A(V , V ) which are standard results in the theory of quantum matrices; their proofs are nothing more than simple computations.
We note that one of the two diagrams that need to commute for V ,V to be a coaction is the diagram in Proposition 2.2 for V = W = U . Therefore we can think of the map V ,W : V → W ⊗ A(W , V ) as a generalization of the coaction. Since the comultiplication is degree preserving (i.e.,

Quantum matrix spaces and e-Hecke pairs
Let V n denote an n-dimensional vector space. Let R n be the R-matrix of U q (gl n ) for the defining representation, namely It is well known that (V n , R n ) form a Yang-Baxter space. Define A q (n, n) := A(V n , V n ).
The space A q (n, n) is the algebra of quantum n × n matrices (see [6,14]). It is a coquasitriangular bialgebra generated by elements x i j , 1 ≤ i, j ≤ n subject to the following RTT relations: We now present some standard properties of A q (n, n), see for example [12,Chapter 7]. We begin by reminding the reader about the coalgebra structure on A q (n, n). The comultiplication and counit are given on generators by The coalgebra structure on A q (n, n) allows one to endow the tensor product V ⊗ W of two A q (n, n)-comodules with the structure of an A q (n, n)-comodule. Therefore the category of finite dimensional A q (n, n)-comodules is a monoidal category. The unit is the trivial comodule k with the coaction 1 ∈ k → 1 ⊗ 1 ∈ k ⊗ A q (n, n). There are standard isomorphisms The bialgebra A q (n, n) is coquasitriangular. This means that there is a map R : A q (n, n)⊗ A q (n, n) → k that is invertible in the convolution algebra, satisfying the following conditions: (2) , b (2) ) = R(a (1) , b (1) )b (2) a (2) for all a, b, c ∈ A q (n, n). In the above formula we use Sweedler notation, namely we denote (a) = a (1) ⊗ a (2) . The map R is given on generators x i j by the formula The values of R on higher order terms is given by repeated applications of the first two equalities in Eq. (7). The existence of R implies that for every A q (n, n)-comodules V , W there is an A q (n, n)comodule isomorphism R V ,W : V ⊗ W → W ⊗ V given by the formula This morphism makes the category of finite dimensional A q (n, n)-comodules into a strict braided monoidal category. A strict braided monoidal category C is a monoidal category with braiding isomorphisms γ V ,W : where I is the identity object in the monoidal category andr V ,l V are the identity constraints in C.

Proposition 2.6
The category of finite dimensional A q (n, n)-comodules is a braided monoidal category with braiding isomorphisms given by When W = V , the map R V := R V ,V satisfies the Yang-Baxter equation. This makes makes the pair (V , R V ) a Yang-Baxter space for every A q (n, n)-comodule V .

Definition 2.7
An e-Hecke pair V is an A q (n, n)-comodule for some n ≥ 1 such that the image of the coaction V : V → V ⊗ A q (n, n) lies in V ⊗ A q (n, n) e .

Remark 2.8
Equivalently, an e-Hecke pair is a (finite dimensional) module over the q-Schur algebra S q (n; e), or equivalently a degree e representation of U q (gl n ). Yet another way to understand the e-Hecke pairs is to note the fact that they are direct sums of subquotients of V ⊗e n where V n is either the n-dimensional defining comodule for A q (n, n) or the defining U q (gl n ) module.
We explain the term "e-Hecke pair". Let V be an e-Hecke pair. First, we call it a "pair" because we think of V as the pair (V , R V ). To explain the word "Hecke", let us start with the case e = 1. If V is indecomposable, then V has to be the defining comodule V n and the action of the braid group B d on V ⊗d n factors through the action of the Hecke algebra H d . This is not the case for general e. Instead, the action of B d on V ⊗d factors through a different deformation of the symmetric group S d , which is realized as a subalgebra of H de as follows. Let w i be the element in S de such that

Definition 2.9
The e-Hecke algebra of rank d, denoted by H d,e , is the subalgebra of H de generated by T w 1 , . . . T w d−1 , where w i are as in Eq. (11).
One sees from the definition that the action of B d on (V ⊗e n ) ⊗d factors through H d,e . Since an indecomposable e-Hecke pair V is a subquotient of V ⊗e n as an A q (n, n)-comodule (see We also have that It follows that the dth tensor power of an indecomposable e-Hecke pair is a module over the e-Hecke algebra H d,e . Note that the last sentence is false if we do not require the e-Hecke pair to be indecomposable, for both e = 1 and general e.

Remark 2.10
Note that the dimension of H d,e is in general greater than that of kS d . When d = 2, we can relate the dimension of H d,e to the eigenvalues of a certain R-matrix, where we can already see the difficulty of computing the dimension. The Schur-Weyl duality identifies H 2e with End U q (gl 2e ) (V ⊗2e 2e ); under this identification, the generating element T w ∈ H 2,e corresponds to the map R V ⊗e 2e . Thus, the dimension of H 2,e is equal to the degree of the minimal polynomial of R V ⊗e 2e . Since any R-matrix is diagonalizable for q generic, the degree is equal to the number of different eigenvalues of the R-matrix R V ⊗e 2e . For example, when e = 2 and q = 1 is generic, the R-matrix R V ⊗2

Remark 2.11
The e-Hecke algebra arises when one considers the wreath product of finite groups. Given two groups G and H ⊆ B d , one can define a wreath product mimicking the usual construction where H ⊆ S d : Let the wreath product G H be the semidirect product G ×d H , where the action of H is the braid group action permuting components. If G = B e and H = B d , then the wreath product B e B d is a subgroup of the larger braid group B de generated by where T i are the standard generators for B de and w i ∈ B de are (unique) shortest lifts of w i ∈ S de in (11). Now we replace the braid groups by Hecke algebras in constructing the (internal) wreath product. The group B e is replaced by the Hecke algebra H e , and the product B ×d e is replaced by H ⊗d e . This latter algebra is a subalgebra of H de generated by The e-Hecke algebra H d,e naturally acts on this; the generator T w i acts as the multiplication in H de . Then, the subalgebra in H de generated by the above H ⊗d e and our e-Hecke algebra H d,e , denoted by H e H d,e , can be thought of as an analog to B e B d or S e S d .
The last equality of Eq. (7) implies that R V is an A q (n, n)-comodule homomorphism. Given an A q (n, n)-comodule V , write the coaction map as The equation above serves as the definition for t V ji .

Lemma 2.12
The equation is equivalent to the fact that R V is an A q (n, n)-comodule homomorphism.
Applying both sides to v i ⊗ v j and picking out the A q (n, n)-coefficients of v p ⊗ v q produces the the desired equation.

Lemma 2.13
The relation holds in A q (n, n).
Proof This follows from Lemma 2.12 and the fact that R V is an A q (n, n)-comodule homomorphism.

Given an
If we define S q (n; d) := S q (V n , V n ; d), then S q (n; d) is the q-Schur algebra due to Dipper and James [4]. Proposition 2.14 Suppose q is generic or 1 and char k = 0. Then the category of finite dimensional A q (n, n)-comodules is semisimple.
Proof See Theorem 11.4.4 in Parshall and Wang [13] where our Proposition is proved for GL q (n). The same result for A q (n, n) follows in a similar way.

Proposition 2.15 Suppose q is generic or 1 and char
Proof This follows from Proposition 2.14 and Remark 2.8.

Definition
In this section we propose a different definition of quantum polynomial functors that generalizes the definition in [10]. Our category of quantum polynomial functors enjoys many of the properties presented in [10], and additionally, it has a composition. Let d, e be non-negative integers.
Let us define the quantum divided power category d q,e V. Its objects are all e-Hecke pairs for all positive n. The morphisms are defined as Note that the category d q,e V always contains a one dimensional e-Hecke pair that is obtained by tensoring the defining A q (1, 1) 1 -comodule with itself e times. It then follows that the category contains an n-dimensional vector space for every positive integer n. When d = 1, the forgetful functor (V , R) → V to finite dimensional vector spaces induces an equivalence of categories 1 q,e V ∼ = V (13) for any q, e. When q = 1, the R-matrix R V of any e-Hecke pair for any d, e, where d V is the domain category for the classical polynomial functors defined in the introduction.

Definition 3.1 A quantum polynomial functor of degree d on e-Hecke pairs is a linear functor
We denote by P d q,e the category of quantum polynomial functors of degree d on e-Hecke pairs. Morphisms are natural transformations of functors.

Definition 3.2 Define the category P •,d
q,e as in Definition 3.1 with the added requirement that the domain consists only of e-Hecke pairs that are a subquotient of V ⊗e n for some n.
Given an object of P d q,e , one can restrict its domain and define an object of P •,d q,e . We will show that the categories P d q,e are the natural setting where one can define composition. The equivalence (14) tells us that if we specialize q = 1 in in both categories defined above, we recover the category of polynomial functors P d of Friedlander-Suslin [7] (see Sect. 1).

Remark 3.3 If
. Therefore the constant functor, mapping an e-Hecke pair V → k, where k is the trivial A q (n, n)-comodule, is a degree 0 polynomial functor on e-Hecke pairs. It's not hard to see all elements in P d q,e are direct sums of the constant functor.
There is an equivalent characterization of a polynomial functor both in the classical and Hong and Yacobi [10] setting, which directly applies to ours.
we have a map which gives rise to two maps

Proposition 3.4 ([10, Proposition 3.5])
A quantum polynomial functor F of degree d is equivalent to the following data: Proof See Proposition 3.5 in [10]. We note that even though they prove it for when V , W , U are defining comodules, their proof goes through unchanged for general e-Hecke pairs as in our setting.
We can extend the map that satisfies the following property.
Proof The diagrams in Proposition 3.4 are exactly what is needed for V F(V ) to be a coaction map.
Let V be an e-Hecke pair. The bialgebra A q (V , V ) is a quotient of the free algebra generated by {x V ji } by the ideal generated by k, Lemma 3. 6 The map ψ V is a bialgebra homomorphim.
Proof We first need to check that ψ V is well defined. We can do this by showing that: The first equality is by definition and the second one holds according to Lemma 2.13. We now show that ψ V is a coalgebra homomorphism. This is equivalent to showing that ψ V commutes with the comultiplication and the counit, namely Both equations follows from the fact that V is a coaction.
Let G be a quantum polynomial functor of degree d. We have maps:

Lemma 3.7 The map
is well-defined. Therefore it is an algebra homomorphism.
Proof The proof is similar to how we proved that ψ V is well-defined in Lemma 3.6. We want The first equality is by definition. The second equality follows from the commutativity of the exterior square in the diagram where m is the multiplication in A q (GW , GV ). The diagram is commutative because the two small squares are: the first by Lemma 2.12 and the second is trivial.

Lemma 3.8
The maps ψ G W ,V satisfy the following two equations for all e-Hecke pairs U , V , W : This in turn follows from the commutativity of diagram (15).
The second equation is equivalent to e A q (V ,V ) (t W ,V i j ) = δ i j which follows from the commutativity of diagram (16).

Basic operations on quantum polynomial functors
We denote by P d q := e P d q,e the category of quantum polynomial functors of degree d, by P q,e := d P d q,e the category of quantum polynomial functors on e-Hecke pairs, and by P q := e,d P d q,e the category of quantum polynomial functors.

Tensor product in P q,e
Given two quantum polynomials on e-Hecke pairs F ∈ P d q,e and G ∈ P d q,e , the (external) tensor product F ⊗ G ∈ P d+d q,e is defined in the same way as in [10]. For an e-Hecke pair V , we define it to be

To define it on the morphisms one uses the inclusion
To be more explicit, for two e-Hecke pairs V , W , the map (F ⊗ G) V ,W is the following composition: Recall the constant functor k ∈ P 0 q,e . It maps an e-Hecke pair V to the trivial A q (n, n)comodule k. It is then an easy exercise using the definition above to show that F ⊗ k ∼ = F via the natural transformation η : F ⊗ k → F given by the standard isomorphism η V : We similarly have that k ⊗ F ∼ = F. This can be summarized as follows: The category P q,e is a monoidal category with the tensor product ⊗ and the unit object k.

Duality in P d q,e
One defines a duality on the functor category using dualities on the domain and codomain categories. In V, we have the linear dual V → V * = Hom k (V , k). For our category d q,e V where the objects have additional structures, we "lift" the linear dual to what is compatible with this structure, namely the twisted dual. To explain the twist here, it is more convenient to work with the U q (gl n )-modules where the twisted duality is rather standard. (Recall that an A q (n, n)-comodule can be thought of as a polynomial representation for U q (gl n ).) For a U q (gl n )-module V , one can define a twisted U q (gl n ) structure on the linear dual V * of the underlying vector space by precomposing the U q (gl n ) action by an antiautomorphism τ 1 (see [11, 9.20] for the definition of τ 1 and the twisted dual). We denote by τ V the resulting U q (gl n )-module. Then we have in U q (gl n )-mod. We remark that τ − is a duality of a highest weight category, under which the irreducibles are self-dual and a standard module and a costandard module of the same highest weight are dual to each other. In particular, the duality preserves the degree of polynomial representations, that is, if V is an U q (gl n )-module of degree e, then the dual τ V is also a degree e U q (gl n )-module, so we can extend this duality on the category of A q (n, n)-comodules. We therefore have a contravariant functor Now we define the duality − # on P d q,e as It is not hard to show that F # satisfies the properties in Proposition 3.4 and therefore it is a quantum polynomial functor. Taking q = 1 in our setting agrees with the classical definition of the dual.

Examples
We present several examples of quantum polynomial functors.

Tensor powers
For each d ∈ N, the dth tensor product functor ⊗ d : V → V ⊗d is a quantum polynomial of degree d.
To be more precise, for each e, there is a functor ⊗ d e in P d q,e which maps an e-Hecke pair V to the de-Hecke pair V ⊗d . The map ⊗ d V ,W on the morphisms is just the inclusion We abusively denote this functor by ⊗ d for any e.
When d = 0 we get the constant polynomial functor ⊗ 0 : V → k, which we denote abusively by k, and ⊗ 1 is the identity functor which we prefer to denote by I .

Divided powers and symmetric powers
Given an e-Hecke pair W , the divided power d,W q,e is the object in P d q,e represented by W , The map on morphisms is given by f → f • −. When W is the e-Hecke pair (k ⊗e , q e 2 ), we denote it by d q,e and call it the divided power. We may drop the index "e" in the notation because it is determined by V . That is, if V is an e-Hecke pair, then d,V q denotes the functor Hom d q,e V (V , −) in P d q,e . Since a divided power is a representable functor, Yoneda's lemma tells us that it is a projective object in P d q,e (see Proposition 6.1 for a detailed proof). Moreover, divided powers form a set of projective generators for P d q,e for generic q as we shall prove in Theorems 6.13 and 6.15.
For an e-Hecke pair V let S d V ∈ P d q,e be defined by where the dual F # of a functor F is defined in §3.2.2. These are the corepresentable objects in P d q,e and form a set of injective cogenerators for P d q,e .

Quantum symmetric, divided and exterior powers
In this subsection we assume char(k) = 0. Let V be an e-Hecke pair. An important set of examples of quantum polynomial functors are the quantum symmetric and exterior powers for generic q due to Berenstein and Zwicknagl [1] (they also require char(k) = 0). These are quantum deformations of the classical symmetric and exterior power, though their theory is significantly more complex. For example, it is not necessary that the dimension of the quantum symmetric power of an U q (gl n )-module to be the same as the dimension of the classical symmetric power of the corresponding U (gl n )-module.
In this subsection we define the quantum symmetric, divided and exterior powers when q is generic or when it is a pth root of unity for an odd integer p. In the generic case, the definition of quantum symmetric and exterior can easily be seen to be the same as in [1].
Let V be an e-Hecke pair. The map R V : V ⊗2 → V ⊗2 is diagonalizable for generic q. For all non-zero q it has eigenvalues ±q r , for a finite number of integers r (see [1] for details; note that they only state it for when V is indecomposable, but it is easy to see that this holds for the direct sum of indecomposables as well). If q is a pth root of unity and p is an odd integer, then q a = −q b for any integers a, b. Therefore the following definition makes sense for this choice of q, even though the R-matrix R V might not be diagonalizable. Define We can now define

Remark 3.10
When q is generic, our quantum divided power agrees with the definition of quantum symmetric power in [1]. Note that semisimplicity makes it unnecessary for [1] to distinguish the two. We prefer to define the quantum symmetric power as a quotient since it is more natural.

Proposition 3.11 The quantum symmetric, exterior and divided powers are quantum polynomial functors.
Proof Let f ∈ Hom B d (V ⊗d , W ⊗d ). In order to show that d q and d q are quantum polynomial functors we need to show that the restriction of f to d q (V ) has image in d q (W ) and the restriction of f to d q (V ) has image in d q (W ). Since f commutes with the action of T i ∈ B d , we have f (V ⊗ · · · ⊗ 2 q (V ) i,i+1 ⊗ · · · ⊗ V ) ⊆ W ⊗ · · · ⊗ 2 q (W ) i,i+1 ⊗ · · · ⊗ W . It then easily follows from the definition that f restricts to a map from d q (V ) to d q (W ). A similar proof works for d q . From the fact that f maps V ⊗· · ·⊗ 2 q (V ) i,i+1 ⊗· · ·⊗V to W ⊗· · ·⊗ 2 q (W ) i,i+1 ⊗· · ·⊗W it follows that f induces a map (which we denote by the same letter) f : S d q (V ) → S d q (W ) from which we deduce that S d q is a quantum polynomial functor.  , and d q all return the classical symmetric power, since in that case q e = 1 = +q r , ∀r .
With our definition, the following statement is clear.

Proposition 3.14 Let V be an e-Hecke pair. Let N be the maximal rank of the Jordan blocks of R V . The sequence
is exact, where p 1 is the inclusion 2 q V − → V ⊗2 (recall that we defined 2 q V as a subspace of V ⊗2 ) and p 2 is defined by Here we record two short exact sequences of degree 2 quantum polynomial functors which is by definition, and which follows from Proposition 3.14. Also note that the first map in (24) and the last map in (25) make an exterior power a direct summand of a tensor power. Furthermore, we have the following.

Proposition 3.15 There are isomorphisms of quantum polynomial functors
Proof Let V be an e-Hecke pair, or equivalently, a polynomial representation of U q (gl n ) of degree e. There is a canonical isomorphism of U q (gl n )-modules, since τ 1 and the comultiplication commute (see [11, 9.20]). We note that what Jantzen denotes by the twisted comultiplication is the usual comultiplication structure that makes A q (n, n) live in the dual of U q (gl n ).) This is then also an isomorphism of A q (n, n)-comodules. By (22), the dual of φ n induces the desired isomorphism ⊗ n ∼ = (⊗ n ) # of polynomial functors.
To prove the other statements, first consider the n = 2 case. The R-matrix of τ V satisfies This follows from the fact that the R-matrix of V * is the transpose of R V (follows easily from Proposition 4.2.7 in [3]) and the matrix of V twisted by τ 1 is also the transpose of R V (follows from the formulas in [11, 9.20]). The isomorphism φ 2 preserves eigenspaces of Rτ V = R V , and hence restricts to an isomorphism from ( 2 q ( τ V )), which is the direct sum of eigenspaces of Rτ V corresponding to eigenvalues −q i , to τ ( 2 q V ), which is the direct sum of eigenspaces of R V corresponding to eigenvalues −q i . Thus, φ 2 induces an isomorphism 2 q ( τ V ) ∼ = τ ( 2 q V ). The duality τ satisfies τ (V ⊗ W ) ∼ = τ V ⊗ τ W because both τ 1 and the antipode are antiautomorphisms. So we have By intersecting we obtain τ ( d q (V )) ∼ = d q ( τ V ), from which n q ∼ = ( n q ) # follows (again, we use (22)).
The second statement is then obtained by comparing the short exact sequence (24) and the dual of the short exact sequence (25) for d = 2. Then it is clear from the definition of the quantum symmetric and divided powers that the same is true for all d.
Remark 3. 16 We now explain why we assume p to be an odd root of unity. Assume q = i = √ −1 and V 2 is the defining A q (2, 2)-comodule. We want to decompose V ⊗2 2 into two parts 2 q (V 2 ) and 2 q (V 2 ). The problem with the construction above is that the R-matrix R 2 only has one eigenvalue q = −q −1 = i. It is not diagonalizable; it has a 2 × 2 Jordan block and two 1 × 1 blocks. Thus, we cannot separate the eigenvalues of R V 2 into "positive" and"negative" eigenvalues and therefore definition (23) doesn't make sense.

Braiding on P q,e
We can use the braiding structure on the category of A q (n, n)-comodules to endow the category (P q,e , ⊗) with the structure of a braided monoidal category.
Let F ∈ P d q,e and G ∈ P d q,e . The tensor products F ⊗ G and G ⊗ F both live in P d+d q,e . For an e-Hecke pair V , recall that F(V ) and G(V ) are both A q (n, n)-comodules for some n according to Proposition 5.
be the braiding isomorphism defined in equation (9). We use it to define the natural transformation This map turns P q,e into a braided monoidal category. But before we can prove that we need two results which are interesting in their own right.
Let V be an e-Hecke Pair and let F ∈ P d q,e and G ∈ P d q,e . Then F(V ) and G(V ) are naturally A q (V , V )-comodules. They also have the structure of A q (n, n)-comodules by Proposition 5.1.
The coalgebra A q (n, n) is coquasitriangular, therefore the comodule structure supplies us with the braiding (7) and is defined on the generators of A q (V , V ) by

Proposition 4.1 The maps R F(V ),G(V ) and R F(V ),G(V ) are equal.
Proof The map ψ V : A q (V , V ) → A q (n, n) defined in equation (17) has the property that for all x, y ∈ A q (V , V ). The equation holds when x, y are generators of i j and therefore the equation above holds when x, y are generators.

Lemma 4.2
The following diagram is commutative: Proof Note that by Proposition 4.1, we can replace in the equation above R F(V ),G(V ) and By considering the commutative diagram in equation (19) in the proof of Lemma 3.7 for the functor F ⊕ G we obtain that the diagram is commutative. The result follows from the fact that the tensor product and that the restriction of

Proposition 4.3 Let F, F ∈ P d q,e be quantum polynomial functors and let α : F → F be a natural transformation. Then the maps α
Proof The following diagram commutes for any f ∈ Hom(V , V ) ∼ = S q (V , V ; d) because α is a natural transformation: This is the same as saying that the map α V is an S q (V , V ; d)-module homomorphism from which the conclusion follows.
Now we can prove the main result of this subsection.

Theorem 4.4 The category P q,e is a braided monoidal category with braiding isomorphism
Proof We first show that the braiding is a well defined morphism in Hom(F ⊗ G, G ⊗ F). This is equivalent to showing the commutativity of the following diagram for all e-Hecke pairs V , W and any f ∈ Hom(V , W ). The commutativity of the diagram above for all f is equivalent to the commutativity of the following diagram where the horizontal maps are given by The commutativity of the second diagram is now equivalent to the commutativity of the following diagram by the definition of S q (V , which is commutative by Lemma 4.2. This completes the proof of naturality of the braiding. Now we show that R F,G is a natural transformation, namely we need to show that for f : F → F and g : G → G we have The relation above holds when applied to any V because we can write R F, )τ and f V ⊗ g V commutes with each factor of that composition except for τ which switches f V and g V by Proposition 4.3. Notice that in the equation above we use R V , the universal R-matrix of the coalgebra A q (V , V ), instead of R. We can do this because of Proposition 4.1.
The natural transformation R F,G is an isomorphism because R F(V ),G(V ) is an isomorphism for every V . We show that it satisfies equations (10). To prove the first property γ V ⊗W ,U = (γ V ,U ⊗ 1)(1 ⊗ γ W ,U ) we note that it is equivalent to for every e-Hecke pair V . This can be rewritten as and follows immediately from Proposition 2.6.
The third propertyr V γ I ,V =l V can be rewritten as which again follows immediately from Proposition 2.6. The rest of the properties properties follow by the same argument as above.

Composition of quantum polynomial functors
Given two linear functors F, G between arbitrary k-linear categories, one can define the composition F • G if the domain of F agrees with the codomain of G, and then F • G is a linear functor. The quantum polynomial functors, as in the definition presented, have (up to equivalence) V as their codomain, which does not match the domains of quantum polynomial functors. We have to endow the image of F with some additional structure.

Proposition 5.1 Given V an e-Hecke pair and F a quantum polynomial functor of degree d, then F(V ) is also an A q (n, n)-comodule. It is a de-Hecke pair.
Proof By Lemma 3.5, This completes the proof of the statement.
We now explain the composition of quantum polynomials functors. Let G ∈ P d 2 q,e and F ∈ P d 1 q,d 2 e . We define F • G ∈ P d 1 d 2 q,e as follows: This composition makes sense because G(V ) is an A q (n, n) d 2 e -comodule by Proposition 5.1. Since F is a quantum polynomial functor of degree d 1 , we have maps that satisfy the commutation relations in Proposition 3.4. G is also a quantum polynomial functor so we have maps: where ψ G V ,W is defined in Lemma 3.7. Theorem 5.2 (F • G) satisfies properties (15), (16) in Proposition 3.4. Therefore F • G is a well-defined quantum polynomial functor in P d 1 d 2 q,e . Proof Diagram (15) for (F • G) is equivalent to the exterior square of the following diagram: We show the commutativity of the exterior square by showing the commutativity of all three interior quadrilaterals. The commutativity of the top left diagram follows from the fact that F is a quantum polynomial functor. The bottom left diagram follows from the fact that the horizontal arrows modify only the left component of the tensor product, while the vertical arrows modify the right component only. The commutativity of the top right diagram follows from the fact that ψ G U ,V satisfies equation (20). Diagram (16) for F • G is the exterior triangle of the following diagram: The left triangle commutes by (16) and the right triangle by (21).

Representability
Recall the divided power functor d,V q,e ∈ P d q,e : Proof We need to show the existence of an isomorphism for any F ∈ P d q,e . Define the map ρ : We now show that ρ and φ are inverses to each other. Start with v ∈ F(W ). We have Now let f ∈ Hom P d q,e ( d,W q,e , F). We want to show that for any U ∈ d q,e and g ∈ Hom B d ((W ) ⊗d , U ⊗d ).
The second equality follows from the commutativity of the following diagram which holds because f is a natural transformation. It follows that d,W q,e is a projective object in P d q,e . Definition 6.2 Let W be an e-Hecke pair. The quantum polynomial functor F ∈ P d q,e is W -generated if for every e-Hecke pair U the map is surjective. We say that the category P d q,e is finitely generated if there is an e-Hecke pair W such that every F ∈ P d q,e is W -generated.
is surjective for any F ∈ P d q,e .
Proof Since there are maps f i , g i such that i g i • f i = Id W ⊗d , by applying F we obtain We obtain an important corollary.
Proof This follows immediately from the proof of Lemma 6.6.

Lemma 6.8 If V generates W then V generates any direct summand of W as an A q (n, n)comodule.
Proof If V generates W and W = W 1 ⊕ W 2 (as A q (n, n)-comodules), then V generates W 1 . To see this, assume the existence of maps f i , g i such that i g i • f i = Id W ⊗d . Since W 1 is a direct summand, there are inclusion and projection maps i : The following lemma is a standard fact in quantum theory. Lemma 6.9 Given a composition λ = (λ 1 , · · · , λ n ) of d, let V q,λ be the subspace of V ⊗d n generated (as a vector space) by the vectors v i σ (1) ⊗ · · · ⊗ v ⊗λ n n , and σ ∈ S d , where {v i } is the standard basis of V n . Then V q,λ is a direct summand of V ⊗d n as a B d -module.
Proof The generator T j of the braid group maps the vector v = · · · ⊗ v i j ⊗ v i j+1 ⊗ · · · to a linear combination of v and v = · · · ⊗ v i j+1 ⊗ v i j ⊗ · · · . See equation (5). It follows that T i leaves V q,λ invariant. Since this is true for all λ, the submodule V q,λ is a direct summand of V d n as B d -modules.
, provided that m ≥ ed. Consider the subgroup, call it B d,e , of B de generated by T w 1 , · · · , T w d−1 , where w i are as in (11).
The action of T w i ∈ B d,e on V ⊗de n is the same as the action of . The B de -maps f i and g i can then be viewed as B d -maps This gives the desired result.
Define (V ⊗e m ) (i) to be a (numbered) copy of V ⊗e m .

Proposition 6.12 The e-Hecke pair ⊕
⊗d is a direct sum of U n,λ over λ. Letλ = (λ 1 , . . .λN ) be the composition λ with all the 0's removed, then N ≤ d. IfN < d, add 0's at the end ofλ such that the number of entries inλ is exactly d.
Define U m,λ ⊂ (⊕ d i=1 (V ⊗e m ) (i) ) ⊗d in much the same way we defined U n,λ above. Then by an argument similar to the proof of Proposition 6.11 it follows that U m,λ generates U n,λ . By Proposition 6.5 we are done.
The special cases in the Propositions 6.11 and 6.12 are enough to guarantee a projective generator in the semisimple situation. The main motivation behind our construction is composition of polynomial functors. For this to work in the greatest generality, we require direct sum of indecomposable e-Hecke pairs to be in the domain. To see this consider the simplest case possible, when e = 1, d = 1. Consider the polynomial functor mapping the 1-Hecke pair V 1 → V 1 ⊕ V 1 . This defines the functor since V 1 is a projective generator by Theorem 6.13. It follows that V 1 ⊕ V 1 must be in the domain if we wish to compose any two polynomial functors (subject to constraints on d and e).
The category P d q,e is equivalent to the module category of a generalized Schur algebra. We now present a category who has the same property, but whose Schur algebra is slightly simpler.
Recall the category P •,d q,e from Definition 3.2. Each quantum polynomial functor in P d q,e can then be restricted to an object of P •,d q,e . The added condition on the domain makes it so that composition is not possible in P •,d q,e (in the simplest case presented above, because V 1 ⊕ V 1 is not part of the domain of P •,d q,1 ). However this condition allows one to prove the existence of a simpler projective generator. Proof The proof follows along the same lines as the proof of Theorem 6.13, only now we require the simpler Proposition 6.11 (instead of Proposition 6.12) because of the extra condition on the domain.

Corollary 6.16 The evaluation functor
is an equivalence of categories for q not a root of unity and char(k) = 0. It follows that S q (V ⊗e n , V ⊗e n , d) and S q (V ⊗e m , V ⊗e m , d) are Morita equivalent when m, n ≥ de.
The category P •,d q,1 is equivalent to the category studied by Hong and Yacobi in [10]. The category P d q,1 is strictly greater. In order to be able to define composition one needs to consider higher degree comodules in the domain (which produces P •,d q,e ) and then also consider direct sums (which produces P d q,e ).

Remark 6.17
Setting e = 1 in Theorem 6.15, we obtain Theorem 4.7 in Hong and Yacobi [10] when q is generic and char(k) = 0. But note that our proof for e = 1 works when q is a root of unity or char(k) = 0 with a minor addition which we now explain. Theorem 6.15 depends on Corollary 6.7, Proposition 6.11 and Lemma 6.8 which are true regardless if q is a root of unity or not and if char(k) is 0 or not. It also depends on Propositions 2.14 and 2.15, which are not true in general for q a root of unity or char(k) = 0. However, when e = 1, Propositions 2.14 and 2.15 hold for q a root of unity or char(k) = 0 because 1-Hecke pairs are just direct sums of the defining comodule V n for any n. Thus we obtain Theorem 4.7 in [10] with no restrictions on q or the characteristic of k.

Remark 6.18
We can think of the generalized Schur algebras S q (V ⊗e n , V ⊗e n ; d) for generic q as follows. Quantum Schur-Weyl duality (due to Jimbo) says that there is a commuting action of the Hecke algebra H d and the q-Schur algebra S q (V n , V n ; d) on the space V ⊗d n and these two actions satisfy a double centralizer property. The e-Hecke algebra H d,e ⊂ H de acts on (V ⊗e n ) ⊗d as explained before. Then S q (V ⊗e n , V ⊗e n ; d) ⊃ S q (V n , V n ; de) is conjecturally the object that makes the following diagram satisfy a double centralizer property on both rows. We now define a category that "lives in between" the category P d q,e for any e and the category P d q . Let d be a positive integer. The quantum divided power category d q V is the category with objects formal finite direct sums ⊕ i V i where each V i is an e-Hecke for some e. The morphisms are defined on homogeneous objects as follows: The Hom extends naturally to all objects via the formula Then it is not hard to see that F e ∈ P d q,e .
We can define a composition on P d q similarly to how we defined composition between a functor in P d 1 q,d 2 e and a functor in P d 2 q,e . It is interesting to note that when q = 1, the category P d q becomes the category of classical polynomial functors P d due to Friedlander and Suslin. Therefore when q = 1, the category P d q and the categories P d q,e for any e are all equivalent. We show that for generic q, P d q is not equivalent to P d q,e for any e. In fact in the quantum case P d q is "closer" to P d q than to P d q,e since both P d q and P d q are not finitely generated while P d q,e is for q is generic. This is another reason why the theory of quantum polynomial functors is richer in the quantum case than it is in the classical case.

Proposition 7.3
Suppose q is not a root of unity. Then the category P d q is not finitely generated. In particular, P d q is not equivalent to P d q,e for any e.
Proof Consider the sequence of objects k ⊗ f in d q V, where f ∈ N and k = (k, q) is the trivial A q (1, 1)-comodule of degree 1. Then each (k ⊗ f ) ⊗d is a one-dimensional B d -module on which each T i acts as multiplication by q l(w i ) = q f 2 . These form an infinite collection of irreducibles B d -modules since q is not a root of unity. Take F = d in 7.1; we show that no V makes the evaluation map surjective for all f . If there was such V , then V ⊗d (as B d -module) would contain all the k ⊗ f above. This is impossible since V is finite dimensional.

Remark 7.4
Note that the proof makes use of the normalization of the R-matrix R n defined in Eq. (5). In particular, we use that The result stays true, while the argument becomes computationally more complicated. One needs to look at R 2 (which will have an eigenvalue −q −2 for the normalization mentioned above) and modify the proof of Proposition 7.3 accordingly.

Remarks on quantum polynomial functors when q is a root of unity
Let q be an lth root of unity where l > 1 is an odd integer.

Representability
The proof of Theorems 6.13 and 6.15 are not valid in this case, since an indecomposable e-Hecke pair W ∈ comod(A q (n, n)) is not necessarily a direct sum of summands of V ⊗e n . But it can still be true that P d q,e is finitely generated, hence equivalent to the module category of a finite dimensional algebra. For example in the classical case (q = 1) when the field k has characteristic p, the category of polynomial functors is not semisimple, but it does have a projective generator just as in the case when the characteristic of the field is 0. In this section we present some remarks on the case when q is a root of unity not equal to 1.
First recall that it is only at the last step of the proof that we use the semisimplicity. In particular, Corollary 6.7, Lemma 6.8 and Proposition 6.5 are valid when q is non-generic. We summarize them as a separate statement.  by B(n i , a i ), where i runs through some finite index set I , n i ∈ N is the rank of the block and a i ∈ k is the generalized eigenvalue of the block. Similarly, name the Jordan blocks of R U by B(m j , b j ), where j ∈ J .
Then the identity map on U ⊗ U factors through V ⊗e n ⊗ V ⊗e n as a B 2 -map if and only if for each j ∈ J , there exists i ∈ I such that a i = b j and n i = m j .
Proof It is enough to factor the identity on each Jordan block of U . But we can do it by embedding B(m j , b j ) onto the block B(n i , a i ).
Therefore understanding the Jordon block decomposition of R U would allow one to prove representability for P •,d q,e . A statement similar to Proposition 8.2, where one replaces the indecomposable U by any e-Hecke pair U can also be proven.

Remark 8.3
The condition in Proposition 8.2 is trivially true if the R-matrix of any e-Hecke pair is diagonalizable. This is the case when q is not a root of unity, but it is not true when q is a root of unity. For example R q is not diagonalizable when q = ±i. If we look only at 1-Hecke pairs, these are the only values for q where R V is not diagonalizable. For a general e-Hecke pair V , we expect there are other roots of unity for which R V is not diagonalizable.

Remark 8.4
It is possible that the functor cohomology for quantum polynomial functors agrees with the corresponding quantum group cohomology even if the category P d q,e does not have a finite generator. A similar approach is found in Suslin's appendix in [5].

An additional structure
An important feature when q is a root of unity is the existence of the Frobenius twist. When e = 1, this structure comes from the q-Schur algebra. Let us write this in terms of polynomial functors. This is explained in a previous version of [10], which we now repeat. The algebra map Fr n,m : A 1 (n, m) → A q (n, m) defined on the standard generators by is also a coalgebra map. Using this map, we can define a functor (−) [1] : P d → P ld q,1 . We call it the Frobenius twist. (We remind the reader that the classical polynomial functor category P d can be viewed as P d 1,e for any e, and that P ld q,1 is equivalent to the category P ld q of Hong-Yacobi [10]. See Remark 6.17.) Given F ∈ P d , its Frobenius twist F [1] is defined to be a functor from d q,1 V to 1 q,d V ∼ = V that sends a 1-Hecke pair V to the vector space F(V ) (forgetting the 1-Hecke structure, V is viewed as an object in d V). To define what F [1] does to morphisms, it is enough to specify the map (F [1] ) V ,W : F(V ) → F(W ) ⊗ A q (W , V ).
Note that V and W are 1-Hecke pairs, hence direct sum of standard A q (n, n)-comodules and A q (m, m)-comodules, respectively, for certain n, m. We define the map Fr : A 1 (W , V ) → A q (W , V ) as a straightforward generalization of Fr n,m . It maps This extends to a bialgebra map. Now we define the map (F [1] ) V ,W ; it is the composition It is not hard to see that (F [1] ) V ,W satisfies the properties needed to make F [1] into a quantum polynomial functor. Consider I [1] ∈ P l q,1 where I ∈ P 1 is the identity polynomial V = 1 V → V. Then for any d, the composition − • − : P d q,l ⊗ P l q,1 → P dl q,1 induces the functor − • I [1] : P d q,l → P dl q,1 .
Since the Frobenius I [1] lives only in the category P l q,1 , we cannot compose the Frobenius with itself multiple times (as in the classical case).
One may ask if we can define a class of objects I [1] e ∈ P l q,e for all e which are reasonable analogues of the Frobenius. This will supply higher Frobenius twists using composition: e := I [1] l r−1 e • · · · • I [1] e .
Precomposing or postcomposing I [r ] e provides functors between various polynomial functor categories.
However, I [1] e does not come from the structure of the q-Schur algebras as in the e = 1 case; if e > 1 we do not have an analogous coalgebra map where V , W are e-Hecke pairs. Thus we must take a different approach to define I [1] e . One may try to define the Frobenius as the cohomology of certain complexes that are of interest by themselves.
That is, one can define the Frobenius polynomial I [1] ∈ P p 1,e as either the kernel or the cokernel of the middle map in (27). Alternatively, the following complex, called the dth de Rham complex, has nontrivial cohomology when p divides d.
If d = ap, its cohomology is given by the Frobenius twist of the (a−1) pth de Rham complex. One can try to quantize these complexes. That gives a way to define I [1] e . Note that all the objects in (27) and (28) except the Frobenius are defined in P d q,e for all e. Also note that the quantum symmetric powers and quantum divided powers usually behave very differently when we move away from the degree 1 case to the degree e case. For example, the dimension of S d q (V ) depends on the A q (n, n)-comodule structure of V and not only on the dimension of V (this phenomenon is investigated in [1]). Therefore the dimension of I [1] e (V ) for an e-Hecke pair V might be different from the dimension of V , in contrary to the classical case. In particular, we cannot obtain I [1] e (V ) from the underlying space of V by just twisting the module structure.
We note that the middle map in the exact sequence (27) can be quantized into a map between polynomial functors acting on 1-Hecke pairs and for q a root of unity (and char(k) = 0) we can define the Frobenius functor as either the kernel or the image of that map. This definition of the Frobenius coincides with definition of I [1] via the Frobenius twist. This a sign that a homological approach to defining the Frobenius in the quantum case is worth investigating.
This discussion can be the starting point of further investigations. One can try to define the quantum Frobenius twist as mentioned above, and try to understand its properties. Then, one can try to understand its role in quantum theory, or its uses in cohomology theory.