Traces and dimensions in braided categories

D. Gurevich, R .Leclercq

ISTV, Université de Valenciennes, 59304 Valenciennes, France

P .Saponov,

Theory Department, Institute for High Energy Physics, Protvino, Russia

April 24, 2001

Abstract

With any even Hecke symmetry R (that is a Hecke type solution of the Yang-Baxter equation) we associate a quasitensor category. We formulate conditions on R under which the category constructed in this way turns out to be a natural generalization of the category of sl(n)-modules. The categorical dimensions for the objects of the category are defined and explicitly calculated. Our main result states that these dimensions are equal to those of U_{q}(sl(n))-modules and can be expressed in terms of the Schur functions. An important property of the approach developed here is its universal character — it does not depend on a specific form of the Hecke symmetry. In particular, it is applicable for the family of even nonquasiclassical Hecke symmetries constructed in [G].

AMS Mathematics Subject Classification, 1991 : 18D10, 81R50 Key words : quasitensor (braided) rigid category, Hecke algebra, categorical dimension

1 Introduction

The main purpose of this paper is to introduce some braided categories arising from ”nonquasiclassical” Hecke symmetries constructed in [G] and to suggest a way of defining and computing dimensions of objects of these categories. Let us precise from the very beginning that by braided categories we mean monoidal tensor or quasitensor ones whose objects are vector spaces with the usual tensor product and whose natural associativity isomorphisms are identical (for the terminology, cf. for example [CP]). Thus, the structure of such a category is principally determined by natural commutativity isomorphisms

R_{U,V} : U ⊗ V → V ⊗ U,  U, V ∈ Ob (C).  (1.1)

The term ”natural” means here that for any two morphisms f : U → U′ and g : V → V′ one has:

(g ⊗ f) ∘ R_{U,V} = R_{U′,V′} ∘ (f ⊗ g).  (1.2)

Here C stands for a category in question and Ob (C) stands for the set of its objects. These isomorphisms are assumed to satisfy the so-called Yang-Baxter equation. They play the role of the usual flip which transposes the factors. We call the isomorphisms (1.1) YB operator, braiding, or quantum R-matrix.

Initially, categories equipped with such morphisms were introduced by S. MacLane [McL1]. However, he only considered involutary YB operators (R^{2} = id ) called in what follows ”symmetries”. A new wave of interest in braided categories (but with noninvolutary braidings) arose in connection with integrable system theory. More precisely, such categories appeared as those of modules of Drinfeld-Jimbo quantum groups (QG) U_{q}(g) playing an important role in this theory (we will denote this category U_{q}(g) − Mod). Besides, the QG have found many other interesting applications, in particular, in noncommutative geometry.

The notion of quantum (or q-) dimension of objects of the category is one of the basic notions related to QG which turns out to be very useful in noncommutative geometry. In accordance with the usual tradition the dimension of a given object V ∈ Ob (C) is the value of the trace operator

tr : End (V) → 𝕂  (1.3)

on the identity element id ∈ End (V). Hereafter 𝕂 stands for the basic field (always ℂ or ℝ).

Let us note that we identify 𝕂 and the End (I) where I is the identity object in the sense of [CP]. Moreover, we use the notation End (V) = Hom (V, V) for the objects of the category which sometimes are called internal (endo)morphisms (cf. [DM]). By contrast, the morphisms of the category in question will be called ”categorical morphisms”. For example, categorical morphisms of the category g − Mod of modules over a Lie algebra g are elements of Hom (U, V), U, V ∈ Ob (g − Mod), commuting with the action of g.

The problem we want to discuss now is a categorical definition of trace (1.3). What is a reasonable generalization of the basic property of the classical trace

tr [X, Y] = 0,  ∀ X, Y ∈ End (V)  (1.4)

where [X, Y] is the commutator of two endomorphisms^{1}?

If the braiding in a given braided category is involutary and invertible by column in the sense of formula (2.47) then there exists a natural generalization of the above commutator such that relation (1.4) is still valid for this commutator and an appropriate trace. For example it is so for a super-commutator and super-trace defined in a super-category. Other examples can be found in [G].

Let us note that the definition of the trace (1.3) in a category equipped with a symmetry makes use of this symmetry and of the identification End (V) = V ⊗ V^{∗} where V^{∗} is an object dual to V (all categories in question are assumed to be rigid, i.e. for any object of the category its dual is also an object). Up to our knowledge for the first time such a trace has been introduced in [DP].

Once such a trace is defined one can introduce a categorical dimension dim(V) of an object V by setting dim(V) = tr (id). It is easy to see that

dim(U ⊕ V) = dim(U) + dim(V), dim(U ⊗ V) = dim(U)dim(V),  (1.5)

i.e. the dimension can be considered as an additive and multiplicative (a-m) functional on the objects of the category.

However, a direct application of the approach of [DP] to the category U_{q}(g) − Mod gives ”dimension” which is not an a-m functional. This is the reason why one has to introduce a correction in the definition of ”dimensions” turning them into an a-m functional. The correction is connected to the so-called ribbon element in the corresponding QG. The image of this element is somewhere called ”twist” (cf. [T]). The categories possessing a braiding and a twist are called ribbon. The trace in such categories is defined via some combination of the ribbon element and another element u due to Drinfeld (cf. [CP]) and this leads to the dimension which is an a-m functional.

Some generalization of this construction of the trace has been suggested in [BW]. Suggested in that paper is a way to define the notion of dimension in some categories without any braiding.

1Another form of this property is tr (g X g^{−1}) = tr X for any invertible g ∈ End (V). Sometimes this property is attributed to the quantum trace (cf. [T]). However, in this case one means by g an invertible categorical morphism (see above). Of course, this property fails if g is an arbitrary (but still invertible) element of End (V).

Such a category is introduced as that of H-modules where H is a Hopf algebra. Instead of a universal quantum R-matrix (i.e. the element giving rise to a braiding) the authors of [BW] use another element allowing to identify any object V with its second dual V^{∗∗}.

In the present paper we restrict ourselves to categories equipped with braidings and suggest another way of introducing traces and dimensions for them.

Before discussing this way let us describe the categories in question. A Yang-Baxter operator

R : V^{⊗2} → V^{⊗2}  (1.6)

will be called a Hecke symmetry if in addition to the Yang-Baxter equation

R_{12} R_{23} R_{12} = R_{23} R_{12} R_{23}  (1.7)

it also satisfies the relation

(q id − R)(q^{−1} id + R) = 0, q ∈ 𝕂.  (1.8)

Here the standard tensor notations have been used for the equations and operators in tensor product of spaces. Equation (1.7) is written in V^{⊗3} and one assumes

R_{12} = R ⊗ id, R_{23} = id ⊗ R,

where id is the identity operator on V.

The nonzero parameter q is taken to be generic, which means it is not a root of unity: ∀k ∈ ℕ, k ≠ 1, q^{k} ≠ 1. As a consequence none of the so-called q-numbers k_{q} is equal to zero

k_{q} ≡ \frac{q^{k} − q^{−k}}{q − q^{−1}} ≠ 0 ∀k ∈ ℕ.  (1.9)

Let us assign to such a Hecke symmetry ”symmetric” Λ_{+}(V) and ”skew-symmetric” Λ_{−}(V) algebras by

Λ_{+}(V) = T (V)/{Im (q id − R)}, Λ_{−}(V) = T (V)/{Im (q^{−1}id + R)}.  (1.10)

Here T (V) stands for the free tensor algebra of the space V and {I} denotes the ideal generated by a subset I in a given algebra. Let Λ_{±}^{l}(V) be a homogeneous component of the algebra Λ_{±}(V) of the degree l. We call a Hecke symmetry even if there exists an integer p such that the component Λ_{−}^{l}(V) is trivial at l > p and it is one-dimensional at l = p. The integer p will be called a rank of V and be denoted rk (V).

In the latter 80’s one of the authors (D.G.) constructed examples of Hecke type braidings which differ drastically from those related to the QG (see [G] and references therein). Namely, it was shown that there exist a lot of Hecke symmetries R such that n = dim(V) > p = rk (V). Here by dim(V) we mean the usual dimension of the space V. Note that these Hecke symmetries are not deformations of the usual flip and we call them nonquasiclassical. To the contrary, the Hecke symmetry coming from the QG U_{q}(sl(n)) is a deformation of the flip and we call it quasiclassical.

Then the following problem naturally arises. Given an even Hecke symmetry, is it possible to associate with it a category looking like U_{q}(g)−Mod and how to define traces and corresponding (categorical) dimensions of its objects? We would like to point out that in what follows we use the term ”dimension” in the categorical sense. The use of classical dimensions will be explicitly specified.

In this paper we introduce a braided category generated by a vector space V equipped with a Hecke symmetry satisfying the additional condition (2.33). This category will be shown to be a natural generalization of the category U_{q}(sl(n)) − Mod. We call it the Schur-Weyl (SW) category and denote as SW(V). Note that for involutary YB operators a similar construction was presented in [GM].

Any object of such a category is a direct sum of simple ones and in this sense the category SW(V) is spanned by simple (basic) objects. These basic objects are labelled by partitions (or what is the same by Young diagrams)

λ = (λ_{1}, λ_{2}, ..., λ_{p−1}), λ_{i+1} ≤ λ_{i},  (1.11)

λ_{i} being nonnegative integers. In the sequel we will use the notation λ ⊢ k for each λ being a partition of the integer k that is ∑_{i} λ_{i} = k. The number of nonzero components of a partition λ is called its height and will be denoted ℓ(λ). For an object corresponding to a diagram λ we introduce the notation V_{λ}.

Let us stress that the tensor product of two basic objects is a direct sum of basic ones. This naturally leads us to the notion of Grothendieck ring of the category which turns out to be the same as for the categories of sl(n)- (or U_{q}(sl(n)))-modules. The only difference is that the role of the classical dimension n = dim(V) is played by the rank p = rk (V). Thus, we can consider the dimensions of objects of the category as a functional on the Grothendieck ring. Since it is an a-m functional we come to the problem of describing the whole family of a-m functionals.

An exhaustive answer to this question is given in Proposition 16 stating that such a functional f is determined by its values on objects V_{(1^{k})}, k = 1, 2, ..., p − 1, and its value on a basic object V_{λ} is equal to

f(V_{λ}) = s_{λ}(α_{1}, α_{2}, ..., α_{p})

where s_{λ} is the Schur function (polynomial) in p variables corresponding to the partition λ. The numbers (−α_{i}) are the roots of the polynomial

ϕ(t) = t^{p} + f_{1}t^{p−1} + … + f_{p−1}t + 1, f_{k} = f(V_{(1^{k})}).

Note, that the classical (usual) dimension is also an a-m functional. Therefore, the above result allows one to calculate the classical dimension of spaces V_{λ} provided that the Poincaré series P_{−}(t) (see Remark 20) is known (for involutary symmetries such a calculation has been done in [GM]). Remark, that if a Hecke symmetry R(q) (1.6) is a deformation of an involutary one R(1) then the usual dimension of corresponding spaces V_{λ} will be the same in both cases. Indeed, being an integer number such a dimension is stable under deformation, while the categorical dimension is a function in q which becomes integer at q = 1.

Having classified all a-m functionals one can put the question: which a-m functionals come from SW categories? More precisely, for which a-m functional on a given Grothendieck ring (with a fixed p) there exists a SW category such that the categorical dimension of its objects coincides with this functional?

Now we are going back to the problem of defining trace on objects End (V). Our definition of trace is rather simple and natural. As a trace we will take a properly normalized (nontrivial) morphism (1.3) in the category. Note that for any object V ∈ Ob (SW(V)) the trivial component in the product V ⊗V^{∗} is unique. So, morphism (1.3) on a given V (being nothing but a multiple of the projection on the trivial component) is uniquely defined up to a (nontrivial) factor and the problem of defining the trace reduces to finding a ”proper” normalization of traces on separate objects (thus, this normalization is a collective phenomenon). Our ”proper” normalization is fixed by the requirement that the corresponding dimension should be an a-m functional.

Having defined the trace and corresponding dimension in a SW category we calculate the categorical dimension dim_{R}(V_{λ}) of basic objects (it suffices for finding dimensions of all objects). The result is rather surprising. The dimension is given by the formula

dim_{q}(V_{λ}) = s_{λ}(q^{p−1}, q^{p−3}, ..., q^{−(p−1)})  (1.12)

which coincides with the well-known Weyl result for the classical dimensions at q = 1. Let us note that in the case of the category U_{q}(sl(n))−Mod formula (1.12) is equivalent to that obtained in [Go]. Thus, the quantum dimension does not depend on which case (quasiclassical or not) we are dealing with. So, we get just the formula for dimensions in the category U_{q}(sl(n)) − Mod with n = p.

However, behind this nondetailed information about the objects (their dimensions) different structures are hidden. For example, traces (1.3) in different categories are defined via some matrix C entering the formula for the trace on the basic space V and extended in a proper way onto the whole category (the extension of C matrix to the space V_{λ} will be denoted C_{λ}). It is worth mentioning that the matrices C_{λ} are essentially different in quasiclassical and nonquasiclassical cases while the categorical dimensions of objects which are nothing but the properly normalized usual traces of C_{λ} depend only on λ and p = rk (V) via formula (1.12).

The paper is organized as follows. In Section 2 we construct all the necessary elements of Schur-Weyl category: the class of objects, categorical morphisms, the quasitensor and rigid structure. In Section 3 we describe the full set of a-m functionals on the objects of SW(V), define the trace in End(U), ∀ U ∈ Ob(SW(V)), and calculate dim_{q} U as the value of trace on identity morphism in End(U). There we give the proof of our main result (1.12).

Acknowledgement Two of the authors (D.G. and P.S) would like to thank Max-Planck-Institut für Mathematik (Bonn) where the paper was completed for warm hospitality. This work was supported in part by the CNRS grant PICS no. 608 and the RFBR grant no. 98-01-22033.

2 Schur-Weyl category generated by a Hecke symmetry

In this section we describe the construction of Schur-Weyl category SW(V) (see Introduction) generated by a finite dimensional vector space V equipped with a Hecke symmetry. We supply the category with attributes of that U_{q}(sl(n))-Mod. In particular, all categories SW(V) are 𝕂-linear rigid quasitensor category (for precise definitions see [CP, McL2]). As was shortly outlined in Introduction, the basic objects of SW(V) are some linear subspaces V_{λ} of the tensor algebra T (V), while the morphisms will be specific linear mappings of the objects (see below). Consequently, the 𝕂-linear structure of such a category is obvious: as the direct sum of objects we will take the usual direct sum of linear spaces and the trivial (zero) vector space will play the role of null object of the category. So we should only define the structure of monoidal category and fix braiding isomorphisms converting the monoidal structure into the quasitensor (or braided) one. Besides, to have a rigid category, we must ensure that the dual space V^{∗} of an arbitrary V ∈ Ob(SW(V)) is also an object of the category. Now we pass to the explicit construction of the mentioned components of the category SW(V).

2.1 Class of objects

Let us fix some finite dimensional vector space V, dim V = n, and choose a basis {e_{i}} in this space. Consider a linear operator R (1.6) whose action on the basis elements reads:

R(e_{i} ⊗ e_{j}) = e_{r} ⊗ e_{s} R_{ij}^{rs},

where the summation over the repeated indices is understood. In what follows we assume R to be a Hecke symmetry with a generic q ∈ 𝕂.

Besides, we will suppose R to be an even symmetry of a rank p ≤ n = dim V. This means that the Poincaré series P_{−}(t) related to the ”skew-symmetric” algebra ∧_{−}(V) (1.10) is a monic polynomial of p-th degree, that is a polynomial in t whose leading coefficient is equal to 1.

Throughout the paper we will use the compact notation for the relations including matrices and vector spaces. In such notations the indices will stand for the number of space rather than for a particular matrix or vector component. A basis vector e_{i} in the k-th matrix space is denoted as e_{⟨k|}, while for the basis vector of dual space we will write e^{|k⟩}. The same principle is applied to components of arbitrary tensors.

For example, the above relation will look like^{2}

R(e_{⟨1|} ⊗ e_{⟨2|}) := e_{⟨1|} ⊗ e_{⟨2|} R_{⟨12|}^{|12⟩} := e_{⟨1|} ⊗ e_{⟨2|} R_{12}.

In this notation the summation over the repeated indices is represented by an expression which contains the same upper and lower case indices with properly oriented brackets as shown in the examples below:

e_{⟨1|} ⊗ e^{|1⟩} := ∑_{i} e_{i} ⊗ e^{i}, e_{⟨1|} · T_{1} := e_{⟨1|} T_{⟨1|}^{|1⟩} := ∑_{j} e_{j}T_{i}^{j}, M_{1}R_{12}N_{2} := ∑_{a,b} M_{a}^{i_{1}}R_{j_{1}b}^{ai_{2}} N_{j_{2}}^{b}

but

e^{|1⟩} ⊗ e_{⟨1|} := e^{i} ⊗ e_{j}, e_{⟨1|} ⊗ e^{|2⟩} := e_{i} ⊗ e^{j}, T_{1} · e_{⟨1|} := T_{j}^{i} e_{k}

and so on.

Now we use the fact, that in each homogeneous component V^{⊗m} of tensor algebra T (V) one can realize the so called local representation of the Hecke algebra H_{m} [DJ] via a given Hecke symmetry^{3}.

The Hecke algebra H_{m} is generated by the unit element id_{H} and m − 1 generators σ_{k} subject to the following relations:

\begin{matrix}σ_{i}σ_{i+1}σ_{i} = σ_{i+1}σ_{i}σ_{i+1} \\ σ_{i}σ_{j} = σ_{j}σ_{i} if |i − j| ≥ 2 \\ (σ_{i} − q id_{H})(σ_{i} + q^{−1} id_{H}) = 0 \end{matrix} \} i = 1, 2, … m − 1.

The local representation of H_{m} in V^{⊗m} is of the form:

σ_{i} → ρ_{R}(σ_{i}) ≡ R_{ii+1} = id_{i−1} ⊗ R ⊗ id_{m−i−1} ∈ End(V^{⊗m}).  (2.1)

In the further construction of SW(V) category the central role belongs to the fact that the Hecke algebra H_{m} for a generic q is semisimple and can be decomposed into a direct sum of ideals. Moreover, since the Hecke algebra H_{m} at a generic q is isomorphic to the group algebra 𝕂[S_{m}] of the m-th order permutation group S_{m}, its primitive idempotents generating the ideals in mentioned decomposition can be put into one-to-one correspondence with the set of all standard Young tableaux connected with each possible partition λ of the integer m. Speaking more explicitly, given the algebra H_{m} and a partition λ ⊢ m, one can construct some polynomials Y_{ii}^{λ} in generators {σ_{i}} which turn out to be the primitive idempotents of Hecke algebra H_{m} (for detailed description of such a construction see review [OP]):

id_{H} = ∑_{λ⊢m} ∑_{i=1}^{d_{λ}} Y_{ii}^{λ},  (2.2)

Y_{ii}^{λ} Y_{jj}^{μ} = δ_{ij}δ^{λμ} Y_{ii}^{λ}.  (2.3)

In the above formulas the index i runs from 1 to d_{λ} ≡ dim λ, that is to the number of all standard Young tableaux, corresponding to the Young diagram of the partition λ ⊢ m. Recall, that a Young tableau is called standard if it is filled with successive integers from 1 to m in such a way that the integers increase from left to right in each row and from top to bottom in each column. The standard Young tableaux (and hence the primitive idempotents Y_{ii}^{λ}) may be lexicographically ordered in many ways and for definiteness we fix an order in which the first idempotent Y_{11}^{λ} corresponds to the Young tableau filled by integers consequently increasing by 1 when going down in each column. Here is an example for the Young tableau corresponding to the partition λ = (3, 2^{2}, 1):

2The symbol := means ”by definition”.

3For the review on Hecke algebra the reader is referred to the recent work [OP]

Y_{11}^{λ} ⟷ 1 5 8 2 6 3 7 4  (2.4)

A primitive idempotent corresponding to the partition λ = (1^{m}) will be called the antisymmetrizer and be denoted as A^{(m)}. There is one important circumstance about the images of A^{(m)} in End(V^{⊗m}) with respect to representation (2.1). Since the YB operator R is taken to be an even Hecke symmetry of rank p, the image of the p-th order antisymmetrizer

A^{(p)} := ρ_{R}(A^{(p)}) : V^{⊗p} → V^{⊗p}

is one dimensional. So, the action of this projector on an arbitrary basis element of V^{⊗p} can be presented in the form:

A^{(p)}e_{⟨1|} ⊗ … ⊗ e_{⟨p|} = e_{⟨1|} ⊗ … ⊗ e_{⟨p|} v^{|1...p⟩}u_{⟨1...p|},  (2.5)

where components of the tensors u and v belong to the field 𝕂. Due to (2.3) the normalization of u and v is fixed to be u_{⟨1...p|}v^{|1...p⟩} = 1. The images of antisymmetrizers A^{(m)} with m > p are trivial:

Im ρ_{R}(A^{(m)}) = 0 ∀m ≥ p + 1.  (2.6)

In accordance with general theory each primitive idempotent Y_{ii}^{λ} generates a left or right ideal of the Hecke algebra H_{m} by means of left or right multiplications on all possible elements of H_{m}. Equivalently, one can consider the left or right regular module over H_{m} which as a vector space is H_{m} itself and left or right action consists in the left or right multiplication by Hecke algebra elements. In these terms the left or right ideals generated by primitive idempotents will be irreducible left or right H_{m} submodules. Let us denote left and right submodules generated by Y_{ii}^{λ} as M_{λ}^{l}(i) and M_{λ}^{r}(i) respectively. As a direct consequence of (2.2) one can write the following decomposition:

H_{m} = ⨁_{λ⊢m} ⨁_{i=1}^{d_{λ}} M_{λ}^{r}(i),  (2.7)

and a similar decomposition holds for left submodules M_{λ}^{l}(i).

Consider in more detail the structure of the component of (2.7) corresponding to a fixed partition λ ⊢ m. Let us denote

M̄_{λ} ≡ ⨁_{i=1}^{d_{λ}} M_{λ}^{r}(i) ≡ ⨁_{i=1}^{d_{λ}} M_{λ}^{l}(i).  (2.8)

The set M̄_{λ} is a two sided submodule in H_{m}. Let us dwell upon the question of linear basis in M̄_{λ}. As is known from the theory of Hecke algebra, M̄_{λ} is isomorphic to the algebra Mat_{d_{λ}}(𝕂) of d_{λ} ×d_{λ} matrices. Therefore in M̄_{λ} one can find the set of d_{λ}^{2} linear independent quantities Y_{ij}^{λ}, which are some polynomials in generators σ_{i} of H_{m} obtained from the primitive idempotents Y_{ii}^{λ}. These quantities form a linear basis in M̄_{λ} and obey the algebra of matrix units^{4} E_{ij} which constitute the linear basis in Mat_{d_{λ}}(𝕂):

Y_{ij}^{λ} Y_{kl}^{λ} = δ_{jk} Y_{il}^{λ}, i, j, k, l = 1 … d_{λ}.  (2.9)

Note, that for each fixed i the elements Y_{ij}^{λ} 1 ≤ j ≤ d_{λ} form a linear basis in the irreducible submodule M_{λ}^{r}(i) ⊂ M̄_{λ} generated by the primitive idempotent Y_{ii}^{λ}. And on the other hand, for each fixed j the elements Y_{ij}^{λ} 1 ≤ i ≤ d_{λ} form a linear basis in M_{λ}^{l}(j) generated by Y_{jj}^{λ}. Therefore we have dim M_{λ}^{r}(i) = dim M_{λ}^{l}(i) = d_{λ}.

Otherwise stated, if we arrange the elements Y_{ij}^{λ} into a rectangular d_{λ} × d_{λ} matrix, then its rows will represent the linear basises of d_{λ} right submodules M_{λ}^{r}(i), while the columns of the matrix will represent the linear basises of left submodules M_{λ}^{l}(i). The diagonal entries of the matrix are the primitive idempotents Y_{ii}^{λ}.

An important property of Y_{ii}^{λ} consists in the following fact. For any two idempotents Y_{ii}^{λ} and Y_{jj}^{λ} corresponding to the same partition λ there exists an invertible element of H_{m} which transforms one of these idempotents into the other one. This means that the submodules M_{λ}^{r}(i) (or M_{λ}^{l}(i)) with different i are isomorphic: they can be transformed into each other by left (respectively right) multiplication on some invertible element of Hecke algebra H_{m}. Note, that the whole M̄_{λ}, being a two-sided submodule in H_{m}, is invariant with respect to such a multiplication.

In what follows we will not distinguish the isomorphic submodules corresponding to a partition λ and will consider them up to an isomorphism. Then each submodule M_{λ}^{r}(i) ⊂ M̄_{λ} generated by Y_{ii}^{λ} can be treated as the image of an irreducible H_{m} module M_{λ}^{r} (labelled only by the partition λ ⊢ m) w.r.t. the following H_{m} module monomorphism:

M_{λ}^{r} → M_{λ}^{r}(i) ⊂ M̄_{λ} ⊂ H_{m}, λ ⊢ m.  (2.10)

In this sense the submodule M̄_{λ} is isomorphic (as the H_{m} module) to a direct sum of d_{λ} copies of the module M_{λ}^{r}:

M̄_{λ} ≡ ⨁_{i=1}^{d_{λ}} M_{λ}^{r}(i) ≅ (M_{λ}^{r})^{⊕d_{λ}}.

This formula reflects the well known fact that the regular representation of a finite dimensional algebra decomposes into a direct sum of irreducible modules and the multiplicity of each module is equal to its dimension.

Remark 1 It is worth mentioning that the module M_{λ}^{r} can be mapped into M̄_{λ} in many different ways and submodules M_{λ}^{r}(i) (generated by a fixed choice of primitive idempotents {Y_{ii}^{λ}}) are only particular cases of all possible monomorphisms. Indeed, let us take an arbitrary set of orthonormal d_{λ} × d_{λ} projectors P^{(i)} of the rank 1:

P^{(i)} · P^{(j)} = δ^{ij} P^{(i)}, ∑_{i=1}^{d_{λ}} P^{(i)} = id .

Then, as is evident from above relations and (2.9), the quantities X_{ii}^{λ} = ∑_{r,s} P_{rs}^{(i)}Y_{rs}^{λ} are also a set of primitive idempotents in H_{m} leading to another decomposition of M̄_{λ} into a direct sum of right submodules which will represent other possible monomorphisms of M_{λ}^{r} into M̄_{λ}.

4Matrix unit E_{ij} 1 ≤ i, j ≤ m of the matrix algebra Mat_{m}(𝕂) is the m × m matrix with the only nonzero entry at the intersection of the i-th row and the j-th column which is equal to 1.

Now with each right^{5} submodule M_{λ}^{r}(i) ⊂ H_{m} we can associate a space V_{λ}(i) ⊂ V^{⊗m} in the following way:

V_{λ}(i) = Im ρ_{R}(M_{λ}^{r}(i)).  (2.11)

Besides, we will deal with a space V̄_{λ} which is the image of M̄_{λ} (2.8):

V̄_{λ} = Im ρ_{R}(M̄_{λ}) ⊂ V^{⊗m}.  (2.12)

The spaces V_{λ}(i) with different i (and all other V_{λ}(X) ∈ V̄_{λ} which are ρ_{R}-images of other possible monomorphisms^{6} of M_{λ}^{r} into M̄_{λ}) are isomorphic as vector spaces and we will not distinguish them. Instead, we will deal with a space V_{λ} which (like M_{λ}^{r}) gives rise to a class of isomorphic embeddings V_{λ} ↪ V̄_{λ} ⊂ V^{⊗m} and each V_{λ}(i) (or any other V_{λ}(X) as well) is just a particular representative of this class of isomorphic spaces.

Remark 2 Formula (2.11) has a nontrivial meaning for k ≥ 1, where at k = 1 the only space V_{λ} is the space V itself. For the future convenience we extend the formula to the case k = 0. Namely, we will take by definition V_{λ⊢0} = V_{0} ≡ 𝕂.

Now we can define the class of objects of the SW(V) category generated by a finite dimensional space V over a field 𝕂 equipped with a Hecke symmetry R (1.6)–(1.8).

Definition 3 To each fixed nonnegative integer k ∈ ℤ_{+} (for k = 0 see remark 2 above) and each possible partition λ ⊢ k we put into correspondence a space V_{λ} isomorphic to any V_{λ}(i) in (2.11). The spaces V_{λ}, λ ⊢ k, k ∈ ℤ_{+} are the basic objects of the category SW(V). The whole class of objects of the category is formed by direct sums of a finite number of basic objects V_{λ}. Thus, the spaces V_{λ}(i) can be treated as the space V_{λ} equipped with an embedding {V_{λ} ↪ V̄_{λ} ⊂ V^{⊗k}}.

Remark 4 The objects V_{λ} should be simple objects of our category. However, some objects V_{λ} of the category will be identified with each other (in particular, V_{(1^{p})} and V_{0}). So, finally simple objects of our category will be V_{λ} modulo the mentioned identification. Up to this identification we will sometimes use the notation [V_{λ}]_{k} for the family of embeddings {V_{λ} ↪ V̄_{λ} ⊂ V^{⊗k}}.

Let us turn now to the definition of morphisms of our category.

2.2 Morphisms of the first kind and the structure of quasitensor category

As a morphism we will take some linear mappings among the objects, which will be described explicitly below.

Definition 5 The morphisms of the first kind are defined as follows :

i) The set Mor(V_{λ}, V_{λ}) for any basic object V_{λ} contains only multiples of the identical morphism. That is for any morphism f : V_{λ} → V_{λ} we have by definition f = a id for some a ∈ 𝕂.

ii) For arbitrary objects of SW(V) category the morphisms of the first kind are represented by a set of linear mappings:

∀ k ∈ ℕ, ∀ τ ∈ H_{k}, ϕ^{k}(τ) = ρ_{R}(τ) : V^{⊗k} → V^{⊗k}.  (2.13)

It can be easily shown, that being restricted to the set of subspaces [V_{λ}]_{k} any mapping ϕ^{k} sends each V_{λ}(i) ∈ [V_{λ}]_{k} to an isomorphic space or to zero space^{7}. This means that nontrivial mappings ϕ^{k} at most change the embedding of V_{λ} into V^{⊗k} and therefore are multiples of identical morphism for the basic objects. Among the morphisms ϕ^{k} there is no one which would send V_{λ} to V_{μ} with λ ≠ μ. A different kind of morphisms is considered in subsection 2.3.

5The choice of right submodules is made since we prefer to use the left action of T ∈ End(V) on V. Since an arbitrary element of M_{λ}^{r}(i) has the form Y_{ii}^{λ}f(σ_{1} … σ_{m}) (f being a polynomial in the generators of H_{m}), then with such a choice the projector ρ_{R}(Y_{ii}^{λ}) will be the last element acting on V^{⊗m} in formula (2.11).

6See Remark 1

7For example, ρ_{R}(σ_{i} + q^{−1} id_{H}) 1 ≤ i ≤ k − 1 sends to zero the image of the antisymmetrizer A^{(k)}.

A similar situation takes place for the category of finite dimensional representations of a simple group G (or a simple algebra 𝔤). Indeed, if one has a linear mapping of an irreducible 𝔤–module into itself such that it commutes with the action of 𝔤 then by the Schur lemma such a mapping equals a id for some a ∈ 𝕂.

The mentioned similarity becomes more obvious if we supply the objects of SW(V) category with the comodule structure over a quantum bialgebra T generated by the well known RT T relation. Namely, consider a 𝕂-algebra generated by n^{2} elements T _{j}^{i} subject to the relations [FRT]:

R_{12}T_{1}T_{2} = T_{1}T_{2}R_{12}.  (2.14)

The bialgebra structure is introduced by a coproduct Δ and a counit ε:

Δ(T _{j}^{i}) = T _{k}^{i} ⊗ T _{j}^{k} ε(T _{j}^{i}) = δ_{j}^{i}.

Define the right comodule structure δ_{r} : V → V ⊗ T on the space V as follows:

δ_{r}(v) = e_{k} ⊗ T _{i}^{k}v^{i}, ∀ v = v^{i}e_{i} ∈ V.

Such a coaction is extended to V^{⊗k} in the obvious way:

δ_{r} : V^{⊗k} → V^{⊗k} ⊗ T e_{⟨1|} ⊗ … ⊗ e_{⟨k|} → e_{⟨1|} ⊗ … ⊗ e_{⟨k|} ⊗ T_{1} … T_{k}  (2.15)

and all the comodule properties are easily verified. Then each V_{λ}(i) λ ⊢ k turns out to be an invariant subcomodule in V^{⊗k} since the coaction δ_{r} commute with the action of ρ_{R}(M_{λ}^{r}(i)) on V^{⊗k}:

∀k, ∀λ ⊢ k, ρ_{R}(M_{λ}^{r}) T_{1}T_{1} … T_{k} = T_{1}T_{1} … T_{k} ρ_{R}(M_{λ}^{r}).

This equation is a direct consequence of defining relations (2.14) and explicit formula for primitive idempotents of H_{k} which expresses each Y_{ii}^{λ} as a polynomial in σ_{1}, σ_{2}, …, σ_{k−1}.

Under certain conditions on matrix R the bialgebra (2.14) can be factorized to a Hopf algebra (see below). In the case of the Hecke symmetry related to the QG U_{q}(sl(n)) this Hopf algebra is in fact the algebra Fun_{q}(SL(n)) — the quantized function algebra on the group SL(n). The dual Hopf algebra (quantum universal enveloping algebra U_{q}(sl(n))) canonically acts on each V_{λ}, therefore we obtain a category of finite dimensional U_{q}(sl(n))–modules. In this case our mappings ϕ^{k} are just those, which commute with the action of U_{q}(sl(n)).

Our next step consists in constructing a monoidal structure, which allows us to ”multiply” the objects of the category. This means that we want to define a covariant functor

⊗ : Ob(SW) × Ob(SW) → Ob(SW)

with some associativity morphisms (for detail see [CP, McL2]). As such a functor we take the usual tensor product of linear spaces with associativity morphisms to be identical. So, we have to prove that the tensor product of any two objects of our category is also an object, that is it can be decomposed into a direct sum of basic objects. Evidently, one only needs to verify this property for the tensor product of basic objects V_{λ}.

Proposition 6 For given λ ⊢ n and μ ⊢ m the tensor product of two basic objects V_{λ} and V_{μ} can be expanded into a direct sum of basic objects V_{ν}, ν ⊢ (n + m):

V_{λ} ⊗ V_{μ} = c_{λμ}^{ν} V_{ν}

the coefficients c_{λμ}^{ν} being the Littlewood-Richardson ones entering the formula for product of Schur symmetric functions s_{λ} in p variables^{8}.

Proof. As is clear from the definition of basic objects V_{λ} of our category the structure of their tensor product is controlled by that of modules M_{λ}^{r} (2.10) and the assertion of the proposition can be reformulated in terms of these modules.

To do so let us consider an embedding of V_{λ} ⊗ V_{μ} into V^{⊗(n+m)} of the form

V_{λ} ⊗ V_{μ} → V_{λ}(i) ⊗ V_{μ}(j) ⊂ V^{⊗(n+m)}

for some fixed i and j. As was defined in (2.11) above, the space V_{λ}(i) is the image of the right H_{n} submodule M_{λ}^{r}(i) under representation ρ_{R} (2.1).

Consider an embedding J : H_{n} × H_{m} ↪ H_{n+m} which on generators looks as follows:

{∀σ′_{i} ∈ H_{n} : J (σ′_{i}) = σ_{i} ∈ H_{n+m} 1 ≤ i ≤ n − 1 \atop ∀σ′′_{j} ∈ H_{m} : J (σ′′_{j}) = σ_{n+j} ∈ H_{n+m} 1 ≤ j ≤ m − 1}  (2.16)

By construction J (H_{n}) and J (H_{m}) form two mutually commuting Hecke subalgebras in H_{m+n}.

Let λ ⊢ n, μ ⊢ m and M′_{λ}(i), M′′_{μ}(j) be two right submodules in H_{n} and H_{m} generated by the primitive idempotents Y_{ii}^{λ}(σ′) and Y_{jj}^{μ}(σ′′) correspondingly. The image of the tensor product M′_{λ}(i) ⊗ M′′_{μ}(j) under (2.16) is a right H_{n} × H_{m} module M_{λ}^{r}(i) ⊗ M_{μ}^{r}(j) ⊂ H_{n+m} and obviously

V_{λ}(i) ⊗ V_{μ}(j) = Im ρ_{R}(M_{λ}^{r}(i) ⊗ M_{μ}^{r}(j)).  (2.17)

The right H_{n+m} module induced from M_{λ}^{r}(i) ⊗ M_{μ}^{r}(j) is reducible and can be decomposed into a direct sum of irreducible right H_{n+m} submodules M_{ν}^{r}(k). This decomposition is the image (under monomorphism M_{λ}^{r} → M_{λ}^{r}(i)) of the following relation between irreducible modules M_{λ}^{r} and M_{μ}^{r}:

M_{λ}^{r} ⊗ M_{μ}^{r} = c_{λμ}^{ν} M_{ν}^{r}  (2.18)

where the coefficients c_{λμ}^{ν} are equal to the multiplicity of irreducible H_{n+m} characters χ^{ν} in the character induced from χ^{λ} × χ^{μ}. Due to (2.17) the coefficients c_{λμ}^{ν} will also determine the expansion of tensor product of two basic objects V_{λ} and V_{μ}.

Now we use the fact that at generic q the Hecke algebra H_{m} is isomorphic to the group algebra 𝕂(S_{m}) of the m-th order permutation group S_{m} for all m ≥ 1. Therefore, the coefficients c_{λμ}^{ν} defining the multiplicity of irreducible module M_{ν}^{r} in the tensor product M_{λ}^{r} ⊗ M_{μ}^{r} are the same for Hecke algebra and for the algebra 𝕂(S_{m}). To complete the proof, note that as is well known from the representation theory of symmetric group (see for example [M]) the corresponding multiplicities coincide with the Littlewood-Richardson coefficients in the product of symmetric Schur functions s_{λ}.

Remark 7 The explicit calculation of the decomposition of M_{λ}^{r}(i) ⊗ M_{μ}^{r}(j) into a direct sum of H_{n+m} submodules gives

M_{λ}^{r}(i) ⊗ M_{μ}^{r}(j) = α_{λμ}^{ν}(k(i, j)) M_{ν}^{r}(k(i, j)), α_{λμ}^{ν}(k(i, j)) ∈ ℤ_{+}.

But the isomorphic submodules M_{ν}^{r}(k) with different k are images of the same module M_{ν}^{r} equipped with different monomorphisms (depending on i and j) M_{ν}^{r} → M̄_{ν} ⊂ H_{n+m}. The sum ∑_{k} α_{λμ}^{ν}(k) does not depend on i and j and is equal to c_{λμ}^{ν} which defines the structure of the tensor product (2.18).

Now we should convert the monoidal category SW(V) into a quasitensor one (see [CP]). This means, that we need to define a set of natural commutativity (or braiding) isomorphisms (1.1) for any pair U, V ∈ Ob(SW(V)). To be compatible with monoidal structure the braiding isomorphisms must satisfy the following property for any triple U, V, W ∈ Ob(SW(V)):

8The detailed description of Schur functions and related topics can be found in [M].

R_{U,V ⊗W} = (id_{V} ⊗ R_{U,W}) ∘ (R_{U,V} ⊗ id_{W}) R_{V ⊗W,U} = (R_{V,U} ⊗ id_{W}) ∘ (id_{V} ⊗ R_{W,U}). (2.19)

Let us recall that the associativity isomorphisms are taken to be identical.

Besides, considering two possible ways of transformation of U ⊗ V ⊗ W into W ⊗ V ⊗ U one comes to the following condition:

(R_{V,W} ⊗id_{U})∘(id_{V} ⊗R_{U,W})∘(R_{U,V} ⊗id_{W}) = (id_{W} ⊗R_{U,V})∘(R_{U,W} ⊗id_{V})∘(id_{U} ⊗R_{V,W}). (2.20)

Let us begin with the simplest object — the space V. As a braiding isomorphism for V ⊗ V we take the Hecke symmetry R. Requirement (2.20) on V^{⊗3} transforms into the Yang-Baxter equation (1.7) on R which is satisfied by the definition of R. The decomposition property (2.19) will be used as a definition of braiding isomorphisms for an arbitrary tensor power of the space V. That is we take:

R_{V,V^{⊗k}} = R_{k}R_{k−1} … R_{1} R_{V^{⊗k},V} = R_{1}R_{2} … R_{k}.  (2.21)

In the above formula a shorthand notation R_{i} ≡ R_{ii+1} is used. Introducing one more notation for the chain of R matrices

R_{i→j} ≡ \{{R_{i}R_{i+1} … R_{j} if j = i + n ≥ i \atop R_{i}R_{i−1} … R_{j} if j = i − n ≤ i}

we can write a compact form for the braiding isomorphism in the general case:

R_{V^{⊗n},V^{⊗m}} = R_{m→(m+n−1)}R_{(m−1)→(m+n−2)} … R_{1→n}
≡ R_{m→1}R_{(m+1)→2} … R_{(n+m−1)→n}.  (2.22)

The two parts of this formula correspond to two possible ways of passing from V^{⊗n} ⊗ V^{⊗m} to V^{⊗m} ⊗ V^{⊗n}. Note, that all mappings (2.22) are morphisms of the first kind.

As for the braiding isomorphisms for 𝕂 ⊗ V we take it to be the usual flip:

𝕂 ⊗ V = V ⊗ 𝕂 = V.  (2.23)

By definition of the objects of the category SW(V) any tensor power V^{⊗m} can be decomposed into a direct sum of basic objects V_{λ}, λ ⊢ m. Therefore by making use of formula (2.22) one can define the braiding isomorphism for the tensor product of two arbitrary basic objects V_{λ} ⊗ V_{μ}, (and, therefore, for the tensor product of any couple of objects) if we manage to prove that isomorphism (2.22) does not ”destroy” the structure of embeddings of V_{λ} ⊗ V_{μ} into V^{⊗n} ⊗ V^{⊗m}. If it is the case then we can take the restriction of (2.22) onto V̄_{λ} ⊗ V̄_{μ} as the braiding morphism for the tensor product of V_{λ} and V_{μ}.

That is we have to show the following. Consider two arbitrary embeddings J_{i} : V_{λ} ↪ V^{⊗n} and J_{j} : V_{μ} ↪ V^{⊗m}. Let J_{i}(V_{λ}) = V_{λ}(i) and J_{j}(V_{μ}) = V_{μ}(j). Let us recall that V_{λ}(i) is defined by (2.11). We want to show that under isomorphism (2.22) one gets:

V_{λ}(i) ⊗ V_{μ}(j) ⟶ R_{μλ}(V_{μ}(j) ⊗ V_{λ}(i))

where R_{λμ} is an invertible operator from End(V̄_{μ} ⊗ V̄_{λ}) (the space V̄_{λ} is defined in (2.12)) which does not depend on i and j.

Proposition 8 For two given partitions λ ⊢ n, μ ⊢ m and two arbitrary integers 1 ≤ i ≤ d_{λ} and 1 ≤ j ≤ d_{μ} consider the corresponding spaces V_{λ}(i) and V_{μ}(j) as defined in (2.11). Then under isomorphism (2.22) one has

V_{λ}(i) ⊗ V_{μ}(j) ⟶ R_{μλ}(V_{μ}(j) ⊗ V_{λ}(i))  (2.24)

where the operator R_{μλ} is defined on V̄_{λ} ⊗ V̄_{μ} by means of the following formula:

R_{μλ} = ρ_{R}(𝕐^{μ}) ⊗ ρ_{R}(𝕐^{λ}) · R_{V^{⊗n},V^{⊗m}} ≡ R_{V^{⊗n},V^{⊗m}} · ρ_{R}(𝕐^{λ}) ⊗ ρ_{R}(𝕐^{μ}).  (2.25)

Here R_{V^{⊗n},V^{⊗m}} is defined by (2.22) and

𝕐^{λ} ≡ ∑_{i=1}^{d_{λ}} Y_{ii}^{λ}

is a central idempotent of the corresponding Hecke algebra.

Proof. The proof is based on the embedding of H_{n} × H_{m} into H_{n+m} introduced in Proposition 6. In accordance with (2.16) and (2.17) we can write

V_{λ}(i) ⊗ V_{μ}(j) = Im{ρ_{R}(Y_{ii}^{λ}) ⊗ ρ_{R}(Y_{jj}^{μ})}.

Considered on the arbitrary basis element of the space V^{⊗n} ⊗ V^{⊗m} the right hand side of the above formula reads:

e_{⟨1|} ⊗ … ⊗ e_{⟨n+m|} → e_{⟨1|} ⊗ … ⊗ e_{⟨n+m|} {Y_{ii}^{λ} ⊗ Y_{jj}^{μ}}_{⟨1...n+m|}^{|1...n+m⟩}  (2.26)

where the matrices Y_{ii}^{λ} = ρ_{R}(Y_{ii}^{λ}) and Y_{jj}^{μ} = ρ_{R}(Y_{jj}^{μ}) are some polynomials in R_{1} … R_{n−1} and R_{n+1} … R_{n+m−1} respectively:

Y_{ii}^{λ} = Y_{ii}^{λ}(R_{1}, …, R_{n−1}), Y_{jj}^{μ} = Y_{jj}^{μ}(R_{n+1}, …, R_{n+m−1}).

Applying the braiding isomorphism (2.22) to V^{⊗n}⊗V^{⊗m} leads to the following result for formula (2.26) :

e_{⟨1|} ⊗ … ⊗ e_{⟨n+m|} → e_{⟨1|} ⊗ … ⊗ e_{⟨n+m|} {R_{V^{⊗n},V^{⊗m}} · Y_{ii}^{λ} ⊗ Y_{jj}^{μ}}

where the symbol · stands for the matrix multiplication.

Now one should take into account the following relations which are direct consequence of (2.22) and Yang-Baxter equation (1.7):

\begin{matrix}R_{V^{⊗n},V^{⊗m}} · R_{i} = R_{i+m} · R_{V^{⊗n},V^{⊗m}} & 1 ≤ i ≤ n − 1 \\ R_{V^{⊗n},V^{⊗m}} · R_{j} = R_{j−n} · R_{V^{⊗n},V^{⊗m}} & n + 1 ≤ j ≤ n + m − 1. \end{matrix}

Using these relations one gets:

R_{V^{⊗n},V^{⊗m}} · Y_{ii}^{λ}(R_{1} … R_{n−1}) ⊗ Y_{jj}^{μ}(R_{n+1} … R_{n+m−1}) =
Y_{jj}^{μ}(R_{1} … R_{m−1}) ⊗ Y_{ii}^{λ}(R_{m+1} … R_{m+n−1}) · R_{V^{⊗n},V^{⊗m}}.

This formula proves (2.24). In order to find the form of R_{μλ} we observe that 𝕐^{λ} = ∑_{i} Y_{ii}^{λ} are central elements of H_{n} (λ ⊢ n) and besides Y_{ii}^{λ} ≡ Y_{ii}^{λ}𝕐^{λ} ∀i, j. Therefore

Y_{jj}^{μ} ⊗ Y_{ii}^{λ} · R_{V^{⊗n},V^{⊗m}} ≡ Y_{jj}^{μ} ⊗ Y_{ii}^{λ} · (ρ_{R}(𝕐^{μ}) ⊗ ρ_{R}(𝕐^{λ}) · R_{V^{⊗n},V^{⊗m}}).

where ρ_{R}(𝕐^{λ}) is the projector onto V̄_{λ} ⊂ V^{⊗(n+m)}. Thus, we come to form (2.25) of R_{μλ}.

So, the operator R_{μλ} = ρ_{R}(𝕐^{μ})⊗ρ_{R}(𝕐^{λ})R_{V^{⊗n},V^{⊗m}} does not depend on a concrete embedding of V_{λ} ⊗ V_{μ} ∈ Ob(SW(V)) into V^{n} ⊗ V^{m} (i.e., it does not depend on indices i, j) and represents a braiding isomorphism for V_{λ} ⊗ V_{μ}.

2.3 Morphisms of the second kind and reduction procedure

As was mentioned above (see (2.6)) the image of the antisymmetrizer A^{m} with m > p is identical zero in V^{⊗m}. It can be shown that the same is true for any Im ρ_{R}(M_{λ}^{r}) in case if the height ℓ(λ) > p. This means that the basic objects of our category V_{λ} are labelled by partitions with restricted height: V_{λ} ≢ 0 ⇔ ℓ(λ) ≤ p.

However, there is another consequence of (2.5) – (2.6) which concerns the objects labelled by partitions with the height equal to p. Consider a partition λ ⊢ (p + m) for some nonnegative integer m such that ℓ(λ) = p. Let μ denote a partition of m which is obtained from λ by striking the first column out of the Young diagram corresponding to λ. The parts of λ and μ are connected by the relation μ_{i} = λ_{i} − 1, ∀λ_{i} ≠ 0. As immediately follows from Proposition 6 and from the above remark about the maximal height of λ the following decomposition takes place:

V_{λ} ≅ V_{(1^{p})} ⊗ V_{μ}  (2.27)

where V_{(1^{p})} stands for the one dimensional space labelled by one-column diagram with p boxes: λ = (1^{p}).

Isomorphisms (2.27) allow us to define the so called reduction procedure and to introduce morphisms of the second kind in our category, namely, those identifying V_{λ} and V_{μ}. For this purpose consider a mapping ψ sending the one-dimensional space V_{(1^{p})} into the field 𝕂. Evidently, one only needs to fix the action of the mapping ψ on a basis vector of V_{(1^{p})} ≅ Im ρ_{R}(A^{(p)}). In the general form such a mapping looks like

ψ(e_{⟨1|} ⊗ … ⊗ e_{⟨p|} v^{|1...p⟩}) = 1 ∈ 𝕂.  (2.28)

The choice of 1 in the above formula does not restrict generality of our consideration.

Let us find conditions under which the mapping ψ can be treated as a morphism in our category. Taking into account (1.2) and the fact that the identity operator is a morphism of the category we conclude that the following diagrams must be commutative:

\begin{matrix}V_{(1^{p})} ⊗ V_{μ} & ⟶^{ψ⊗id} & 𝕂 ⊗ V_{μ} \\ ^{R_{V_{(1^{p})},Vμ}} ↓ & & ∥ \\ V_{μ} ⊗ V_{(1^{p})} & ⟶^{id⊗ψ} & V_{μ} ⊗ 𝕂 \end{matrix}  \begin{matrix}V_{μ} ⊗ V_{(1^{p})} & ⟶^{id⊗ψ} & V_{μ} ⊗ 𝕂 \\ ^{R_{Vμ,V_{(1^{p})}}} ↓ & & ∥ \\ V_{(1^{p})} ⊗ V_{μ} & ⟶^{ψ⊗id} & 𝕂 ⊗ V_{μ} \end{matrix}

Let us consider a particular case of these diagrams by putting V_{μ} = V (if they are commutative in this case the same will be true for any μ). Then by passing to the basis we have

V_{(1^{p})} ⊗ V : e_{⟨1|} ⊗ … ⊗ e_{⟨p|} ⊗ e_{⟨p+1|} v^{|1...p⟩} ⟶^{ψ⊗id} 1 ⊗ e_{⟨p+1|} = e_{⟨p+1|} ⊗ 1.  (2.29)

According to another way in the left diagram we should first apply the braiding morphism R_{V_{(1^{p})},V}. Thus, we have

R_{V_{(1^{p})},V} : e_{⟨1|} ⊗ … ⊗ e_{⟨p|} ⊗ e_{⟨p+1|} v^{|1...p⟩} ⟶
e_{⟨1|} ⊗ … ⊗ e_{⟨p|} ⊗ e_{⟨p+1|} R_{1} … R_{p} v^{|1...p⟩}.

In order to simplify this expression we need the following useful relations (see [HIOPT]):

R_{1} … R_{p} A^{(p)} = (−1)^{p−1} qp_{q} A^{(2,p+1)}A^{(p)}
R_{p} … R_{1} A^{(2,p+1)} = (−1)^{p−1} qp_{q} A^{(p)}A^{(2,p+1)}  (2.30)
A^{(p)}A^{(2,p+1)}A^{(p)} = p_{q}^{−2}A^{(p)} ⊗ id_{p+1}.

In these formulas the antisymmetrizer A^{(2,p+1)} has the same form as A^{(p)} but it depends on R_{2}, …, R_{p} instead of R_{1}, …, R_{p−1}.

Now in virtue of (2.30) we obtain:

R_{1} … R_{p} v^{|1...p⟩} = v^{|2...p+1⟩} N_{⟨p+1|}^{|1⟩}  (2.31)

where the matrix N is defined as follows:

N_{j}^{i} ≡ (−1)^{p−1}qp_{q} u_{a_{2}...a_{p}j}v^{ia_{2}...a_{p}} or N_{⟨1′|}^{|1⟩} = (−1)^{p−1}qp_{q} u_{⟨2...p1′|}v^{|12...p⟩}.  (2.32)

Therefore the composition (id ⊗ ψ) ∘ R_{V_{(1^{p})},V} gives the following result:

e_{⟨1|} ⊗ … ⊗ e_{⟨p|} ⊗ e_{⟨p+1|} v^{|1...p⟩} → e_{⟨1|} ⊗ … ⊗ e_{⟨p|} ⊗ e_{⟨p+1|} v^{|2...p+1⟩} N_{⟨p+1|}^{|1⟩} → e_{⟨1|}N_{⟨p+1|}^{|1⟩} ⊗ 1

which obviously differs from (2.29). These results can be made compatible iff N is a scalar matrix (i.e., it is proportional to the n × n unit matrix I). If it is so, then by multiplying the braiding R by an appropriate factor we can achieve the commutativity of the left diagram above. Indeed, if we replace the braiding R by aR, a ∈ 𝕂, the factor a^{p} appears in the right hand side of (2.31). Choosing properly the factor a we can obtain a^{p}N = I.

Applying the same procedure for the space V ⊗ V_{(1^{p})} (the right diagram above) we find:

e_{⟨1|} ⊗ e_{⟨2|} ⊗ … ⊗ e_{⟨p+1|} v^{|2...p+1⟩} ⟶ 1 ⊗ e_{⟨p+1|}M_{⟨1|}^{|p+1⟩}

where^{9}

M_{j}^{i} ≡ (−1)^{p−1}qp_{q} u_{ja_{2}...a_{p}}v^{a_{2}...a_{p}i} or M_{⟨1|}^{|1′⟩} ≡ (−1)^{p−1}qp_{q} u_{⟨12...p|}v^{|2...p1′⟩}.  (2.32′)

For the same reason we assume the matrix M to be scalar. Then the right diagram can be also made commutative by a proper rescaling of R. However if we want simultaneously obtain the unit matrix instead of the matrices N and M we have to impose one more condition: M = N.

Therefore from now on we will suppose the Hecke symmetry R to satisfy the relation

M = N = aI, a ∈ 𝕂^{×}.  (2.33)

Thus, the mapping ψ becomes a morphism of the category SW(V) after a proper renormalization of the braiding morphisms.

This normalization factor is easy to find. Indeed, using relations (2.30) (valid for any Hecke symmetry of the rank p) we find

M · N = q^{2}I.  (2.34)

So, if relation (2.33) is satisfied then by virtue of (2.34) we have a = ±q and hence

N = M = ±qI.  (2.35)

Therefore to convert ψ into a morphism we should pass from the initial Hecke symmetry R to the braiding

R̄ = (±q)^{− \frac{1}{p}} R.  (2.36)

On the higher tensor powers this renormalization gives rise to a renormalized braiding as well

R̄_{V^{⊗n},V^{⊗m}} = (±q)^{− \frac{mn}{p}} R_{V^{⊗n},V^{⊗m}}.  (2.37)

9Note that the matrices N and M differ by a factor from those considered in [G].

The mapping ψ (as well as its inverse ψ^{−1}) will be called a morphism of the second kind.

The sense of restriction (2.33) becomes more clear if objects of our category are supplied with a comodule structure over bialgebra (2.14). Consider shortly the consequences of this structure.

Let us define a quantum determinant det_{q} T of bialgebra (2.14) as follows [FRT]:

det_{q}T ≡ u_{⟨12...p|}T_{1}T_{2} … T_{p}v^{|12...p⟩}.  (2.38)

The commutation relations of det_{q} T with generators of (2.14) read (cf. [G]):

det_{q}T · T = (N^{−1}T N) · det_{q}T ⇔ det_{q}T · (NT) = (T N) · det_{q}T  (2.39)

or due to (2.34)

det_{q}T · T = (MT M^{−1}) · det_{q}T ⇔ det_{q}T · (T M) = (MT) · det_{q}T.

If we want the mapping ψ (2.28) to commute with the comodule structure we have to introduce one more condition:

det_{q}T = 1.

But such a relation is compatible with the algebraic structure only if det_{q} T is a central element of algebra (2.14). By virtue of (2.39) this means that the matrix N (and hence M) must be scalar.

Remark 9 However, it is not clear whether there exist Hecke symmetries such that the matrices N and M are scalar but not equal to each other. (Conjecturally, the condition ”N and M are scalar” leads to ”N and M are equal”.) Nevertheless, the family of even Hecke symmetries satisfying (2.33) is sufficiently large. An explicit construction of such a type Hecke symmetries will be given by one of the authors (R.L) in a separate paper.

Let us also remark that for the quasiclassical Hecke symmetries appearing from the universal R matrix of QG U_{q}(sl(n)) condition (2.33) holds true. The quantum determinant det_{q} T and the unit element 1 generate the center of (2.14) [FRT]. But in general (even in the quasiclassical case) det_{q} T is not central.

2.4 The structure of rigid category

To convert a (quasi)tensor category C into a rigid one we must specify the following data [CP]:

a. A mapping ∗ : Ob(C) → Ob(C) which for any U ∈ Ob(C) put into correspondence its left dual U^{∗}

b. For any pair of dual objects U and U^{∗} there exist evaluation ev_{U} and coevaluation π_{U}

morphisms of C

π_{U} : 𝕂 ⟶ U ⊗ U^{∗}, ev_{U} : U^{∗} ⊗ U ⟶ 𝕂

such that the following diagrams are commutative:

U ⟶^{π_{U} ⊗id} (U ⊗ U^{∗}) ⊗ U id↓  ↓id U ⟵^{id⊗ev_{U}} U ⊗ (U^{∗} ⊗ U)  U^{∗} ⟶^{id⊗π_{U}} U^{∗} ⊗ (U ⊗ U^{∗}) id↓  ↓id U^{∗} ⟵^{ev_{U} ⊗id} (U^{∗} ⊗ U) ⊗ U^{∗}  (2.40)

Let us begin with the case U = V. By our construction of the morphism ψ it is evident that the dual space V^{∗} is nothing but Λ^{p−1}(V). So, in order to satisfy (2.40) we should only fix a convenient basis converting Λ^{p−1}(V) into a left dual space.

Definition 10 The space Λ^{p−1}(V) equipped with the basis

e^{i} = e_{a_{2}} ⊗ … ⊗ e_{a_{p}} v^{ia_{2}...a_{p}} or e^{|1⟩} = e_{⟨2|} ⊗ … ⊗ e_{⟨p|} v^{|12...p⟩}.  (2.41)

will be called a left dual space to the space V. The morphisms ev_{V} and π_{V} are defined as follows (on basis vectors):

\begin{aligned}ev_{V} & : & e^{i} ⊗ e_{j} → δ_{i}^{i} 1 \\ π_{V} & : & 1 ⟶ e_{i} ⊗ e^{i}. \end{aligned}  (2.42)

In order to justify this definition we have to show the following.

Proposition 11 Mappings (2.42) are morphisms of the SW(V) category and they satisfy property (2.40).

Proof. The fact that π_{V} is a morphism is evident since

π_{V} = ψ^{−1}.

To show that ev_{V} is a morphism let us consider the mapping

φ = (−1)^{p−1}p_{q}ψ ∘ A^{p} : V^{⊗p} → 𝕂.

It is a morphism by construction. In a basis form we get

φ : e_{i_{1}} ⊗ ... ⊗ e_{i_{p}} → (−1)^{p−1}p_{q}u_{i_{1}...i_{p}}.

Applying this morphism to the element

e^{i} ⊗ e_{j} = v^{ia_{2}...a_{p}}e_{a_{2}} ⊗ ... ⊗ e_{a_{p}} ⊗ e_{j}

we get

φ(e^{i} ⊗ e_{i}) = (−1)^{p−1}p_{q}v^{ia_{2}...a_{p}}u_{a_{2}...a_{p}j} = q^{−1}N_{j}^{i} = ±δ_{j}^{i}.

Thus, the mapping ev_{V} is nothing but the morphism ±φ restricted on Λ^{p−1}(V) ⊗ V. Diagrams (2.40) are obviously commutative. Let us define now left duals to other simple objects of our category. First, introduce a dual

object to V^{⊗m} by putting (V^{⊗m})^{∗} = (V^{∗})^{⊗m}.

Proposition 12 The space (V^{∗})^{⊗m} is dual to the space (V^{⊗m}) being equipped with the mappings

ev_{U} : V^{⊗m} ⊗ V^{∗}^{⊗m} ⟶ 𝕂, π_{U} : 𝕂 ⟶ V^{∗}^{⊗m} ⊗ V^{⊗m}

which are defined to be:

\begin{aligned}ev_{U} & : & e^{|m⟩} ⊗ … ⊗ e^{|1⟩} ⊗ e_{⟨1|} ⊗ … e_{⟨m|} → δ_{⟨1|}^{|1⟩} … δ_{⟨m|}^{|m⟩} \\ π_{U} & : & 1_{𝕂} → e_{⟨1|} ⊗ … ⊗ e_{⟨m|} ⊗ e^{|m⟩} ⊗ … ⊗ e^{|1⟩} \end{aligned} .  (2.43)

Proof is obvious.

Now we are ready to define the dual V_{λ}^{∗} and corresponding morphisms π_{V_{λ}} and ev_{V_{λ}} for arbitrary basic object V_{λ} of the category SW(V).

Fix a basic object V_{λ}, λ ⊢ m, and consider its particular embedding into V^{⊗m} in the form (2.11):

V_{λ}(i) = Im ρ_{R}(Y_{ii}^{λ}).

This space is spanned by the following vectors

e_{⟨1|} ⊗ … ⊗ e_{⟨m|} · (Y_{ii}^{λ})_{⟨1...m|}^{|1...m⟩}

where Y_{ii}^{λ} is a matrix of Y_{ii}^{λ} in the representation ρ_{R}.

Let us define a space V_{λ}^{∗}(i) ⊂ V^{∗}^{⊗m} as a linear span of the form:

V_{λ}^{∗}(i) = {w^{∗} : w^{∗} = w_{⟨1...m|} (Y_{ii}^{λ}) · e^{|m⟩} ⊗ … ⊗ e^{|1⟩}}  (2.44)

where w_{⟨1...m|} is an arbitrary tensor with components from 𝕂. It is important, that the order of spaces in this formula is reversed comparing with the preceding relation while the matrix Y_{ii}^{λ} is the same.

Finally, we have the following.

Proposition 13 The space V_{λ}^{∗} being equipped with the morphisms ev_{V_{λ}} and π_{V_{λ}} of the form

\begin{aligned}ev_{V_{λ}} & : & V_{λ}^{∗} ⊗ V_{λ} ⟶ Y_{ii}^{λ} \\ π_{V_{λ}} & : & 1_{𝕂} ⟶ e_{⟨1|} ⊗ … e_{⟨m|} ⊗ Y_{ii}^{λ} · e^{|m⟩} ⊗ … ⊗ e^{|1⟩}. \end{aligned}  (2.45)

becomes left dual to the space V_{λ} (provided that V_{λ} is realized as V_{λ}(i)).

Let us note that in fact the space V_{λ}^{∗} is nothing but

V_{μ}, μ = (λ_{1}, λ_{1} − λ_{p−1}, λ_{1} − λ_{p−2}, ..., λ_{1} − λ_{2})

equipped with an embedding in the space V^{∗}^{⊗m}. This shows that we do not need to construct left dual to the space V_{λ}^{∗} because it is just the object V_{λ}.

Nevertheless, the problem of an explicit pairing

V_{λ} ⊗ V_{λ}^{∗} → 𝕂

converting V_{λ}^{∗} into the right dual to V_{λ} (and V_{λ} in the left dual to V_{λ}^{∗}) is very important. In order to describe this pairing in a particular case V_{λ} = V let us calculate the braiding of V and Λ^{p−1}(V) expressed via the basis {e^{i}}. Note, that we use the renormalized braiding R̄ (2.36).

Proposition 14 We have

R̄_{V,V^{∗}} : e_{i} ⊗ e^{j} ⟶ e^{r} ⊗ e_{s} Q̄_{ir}^{js}  (2.46)

where in the given basis {e_{i}} the operator Q̄ satisfies the relation:

R̄_{jb}^{ia} Q̄_{al}^{bk} = δ_{l}^{i}δ_{j}^{k} ⇔ Q̄_{jb}^{ia} R̄_{al}^{bk} = δ_{l}^{i}δ_{j}^{k}.  (2.47)

Note, that if such Q̄ exists then the corresponding braiding R̄ is usually called invertible by column.

Proof. According to definition (2.41) for e^{i} and (2.37) for the renormalized braiding we get (omitting the obvious signs of tensor product):

e_{i} ⊗ e^{j} = e_{i} e_{a_{2}} … e_{a_{p}} v^{ja_{2}...a_{p}} ⟶^{R̄_{V,V^{⊗(p−1)}}} e_{s_{1}} … e_{s_{p}}(±q)^{\frac{1−p}{p}} (R_{p−1} … R_{1})_{ia_{2}...a_{p}}^{s_{1}...s_{p}}v^{ja_{2}...a_{p}}.

Now from (2.30) it is easy to find

(±q)^{\frac{1−p}{p}} (R_{p−1} … R_{1})_{ia_{2}...a_{p}}^{s_{1}...s_{p}}v^{ja_{2}...a_{p}} = Q̄_{ir}^{js_{p}} v^{rs_{1}...s_{p−1}}.

Substituting this into the previous relation and applying again the definition (2.41) of the dual basis we find result (2.46).

Now we will introduce a pairing between V and V^{∗} by putting

ēv_{V} = ev_{V} ∘ R_{V,V^{∗}} : V ⊗ V^{∗} → 𝕂.

It is this operator which plays the central role in the next section where a categorical trace in the space End (V) will be introduced.

3 Additive-multiplicive functional and dimension

In this section we find all possible additive and multiplicative (a-m) functionals on objects of Schur-Weyl category SW(V) and suggest a definition of a trace in End(V) as morphism (1.3) in the category such that the corresponding dimension is an a-m functional. As was pointed out in the Introduction, the requirement that the trace should be a morphism in general is not a trivial one. In particular, this requirement gives rise to the notion of quantum trace in the category of finite dimensional modules over a special class of quasitriangular Hopf algebras — the so called ribbon Hopf algebras. In our approach the morphism property of trace can be obtained without any additional Hopf structure.

Let us recall that a mapping f : Ob(SW(V)) → 𝕂 will be called an additive-multiplicative functional (a-m functional) if it possesses the following property:

∀ U, V ∈ Ob(SW(V)) : f(U ⊕ V) = f(U) + f(V), f(U ⊗ V) = f(U)f(V).  (3.1)

Remark 15 Introducing the Grothendieck ring of the category SW(V) and making use of the fact that our category is semisimple (i.e. any its object can be decomposed into a direct sum of simple ones) we conclude that each a-m functional on the category is in fact defined on the Grothendieck ring.

The fact that we are working with a Hecke symmetry of the rank p leads to an important consequence for the set of a-m functionals. Namely we will show that the value of any a-m functional f on a basic object V_{λ} (and, hence, on an arbitrary object of SW(V)) is completely determined by (p − 1) numbers^{10} f(V_{(1^{k})}) k = 1, …, p − 1.

Proposition 16 Given an a-m functional f, construct a p-th order polynomial in a formal variable t of the form:

ϕ(t) = t^{p} + f_{1}t^{p−1} … + f_{p−1}t + 1, f_{k} ≡ f(V_{(1^{k})}), k = 1, …, p − 1.  (3.2)

Let the numbers (−α_{i}) ∈ 𝕂 be the roots of ϕ(t), that is ϕ(−α_{i}) = 0 i = 1, …, p. Then for any basic object V_{λ} of SW(V) the following relation holds:

f(V_{λ}) = s_{λ}(α_{1}, α_{2}, …, α_{p})  (3.3)

where s_{λ}(x_{1}, …, x_{p}) is a symmetric Schur function in p variables^{11} corresponding to the partition λ.

Proof. First of all let us prove that the numbers f_{k} = f(V_{(1^{k})}) do satisfy (3.3). In accordance with the definition of α_{i} we can write

ϕ(t) = (t + α_{1})(t + α_{2}) … (t + α_{p}).

Therefore, as immediately follows from this relation, the coefficient f_{k} is the k-th elementary symmetric function e_{k} in variables α_{i} [M]:

f_{k} = ∑_{i_{1}<i_{2}<...<i_{k}} α_{i_{1}}α_{i_{2}} … α_{i_{k}} = e_{k}(α_{1}, …, α_{k}).

But since s_{(1^{k})} = e_{k} we conclude that the assertion of the proposition is valid for f_{k}.

The fact that the quantities s_{λ}(α_{1}, α_{2}, …, α_{p}) can be considered as values of an a-m functional follows immediately from the properties of Schur functions and from Proposition 6. In order to show the relation (3.3) we should only check that once the quantities f(V_{λ}), λ = (1^{k}), are given then all quantities f(V_{μ}) for all other partitions are uniquely defined. This can be shown by induction in couples (m, k) where m is the number of columns in λ and k is the number of boxes in the last column. Namely, using the implication

10 Due to (2.23) and reduction morphism ψ (2.28) we have to put for any a-m functional f(𝕂) = f(V_{(1^{p})}) = 1.

11Let us recall, that due to (2.23) all the partitions λ labelling the objects of our category has the height not greater than p therefore the right hand side of equation (3.3) is correctly defined for any object of SW(V).

f(V_{λ} ⊗ V_{μ}) = f(V_{λ})f(V_{μ}) = c_{λμ}^{ν} f(V_{ν}) ⇒ s_{λ}s_{μ} = c_{λμ}^{ν} f(V_{ν})

we can find the quantity f(V_{λ}) for λ corresponding to a given couple (m, k) provided that all f(V_{μ}) where μ corresponds to the couples (l, r) such that l < m or l = m, r < k are already known.

In what follows we will use this proposition for a proper definition of the categorical trace. It will be defined as such a morphism (1.3) that the corresponding dimension becomes an am functional. By virtue of the above proposition we should only find the quantities (3.3) for λ = (1^{k}).

Describe now this construction in detail. Begin with the basic object V. Since the space V is finite dimensional, we can identify End(V) and V ⊗ V^{∗} in the usual way:

∀ F ∈ End(V) ⟷ e_{i} ⊗ F_{j}^{i}e^{j} ∈ V ⊗ V^{∗}

F_{j}^{i} being a matrix of F in the basis {e_{i}}. Define the mapping tr_{V} : End(V) → 𝕂 as the following composition of morphisms:

tr_{V} = α ev_{V} ∘ R̄_{V,V^{∗}}.  (3.4)

Note, that tr_{V} is defined by the above relation up to an arbitrary nonzero factor α which will be specified later from the requirement that such a trace would be an a-m functional. In the fixed basis {e_{i}} of V one can write this mapping in an explicit form. Taking R̄_{V,V^{∗}} from Proposition 14 we find:

∀F ∈ End(V) : e_{i} → e_{j}F_{i}^{j} ⇒ tr_{V} (F) = α Tr(F · C)  (3.5)

where the symbol Tr means the usual matrix trace and the n × n matrix C is defined to be

C_{j}^{i} := ∑_{a=1}^{n} Q_{ja}^{ia}, Q ≡ (±q)^{− \frac{1}{p}} Q̄.

Here the matrix Q̄ is taken from (2.47) and the factor (±q)^{\frac{1}{p}} is included into α.

The matrix C has the following useful properties [GPS]:

R_{12}C_{1}C_{2} = C_{1}C_{2}R_{12}  (3.6)
C_{⟨1′|}^{|1⟩} = \frac{p_{q}}{q^{p}} u_{⟨1′23...p|}v^{|123...p⟩} ⇒ Tr C = \frac{p_{q}}{q^{p}} .  (3.7)
Tr_{(2)}R_{12}C_{2} = id_{(1)}.  (3.8)

The trace tr_{V^{⊗k}} in End(V^{⊗k}) is defined (also up to an arbitrary factor β_{(k)}) as the mapping V^{⊗k} ⊗ V^{∗⊗k} → 𝕂:

tr_{V^{⊗k}} = β_{(k)} tr_{V}^{(1)} ∘ tr_{V}^{(2)} … ∘ tr_{V}^{(k)}  (3.9)

where tr_{V}^{(i)} : V^{⊗i} ⊗ V^{∗⊗i} → V^{⊗(i−1)} ⊗ V^{∗⊗(i−1)} reads

tr_{V}^{(i)} = id^{⊗(i−1)} ⊗ α^{−1} tr_{V} ⊗ id^{⊗(i−1)}.

For any basic object V_{λ} λ ⊢ m one can find the explicit form of the trace in End(V_{λ}) ≅ End(V_{λ}(i)) using definitions (2.11), (2.44) and property (3.6):

∀ F ∈ End(V_{λ}(i)) : tr_{V_{λ}}(F) = β_{(m)} Tr_{(1...m)}(F · C_{λ})  (3.10)

where Tr_{(1...m)} is the the usual trace in the matrix space Mat_{n×n}^{⊗m}(𝕂) and the matrix C_{λ} is of the form:

C_{λ} = Y_{ii}^{λ}(R_{1} … R_{m−1})C_{1}C_{2} … C_{m}.  (3.11)

Remark 17 At the first sight the trace tr_{V_{λ}} depends on a concrete embedding V_{λ} → V_{λ}(i). But in fact our definition is invariant w.r.t. different embeddings of V_{λ} into V^{⊗m}. Indeed, as was already noticed in the previous section, the primitive idempotents Y_{ii}^{λ} of the Hecke algebra H_{m} can be transformed into any other idempotent Y_{jj}^{λ} (with the same λ) with the help of an invertible element X(i, j|λ) from H_{m}. The image X = ρ_{R}(X(i, j|λ)) of the element X(i, j|λ) under the representation ρ_{R} (2.1) represents a first kind morphism of SW(V) and according to (3.6) the string of matrices C_{1} … C_{m} in (3.11) commute with X. Therefore trace (3.10) is actually independent of the index i labelling the concrete embedding V_{λ} → V_{λ}(i).

Find now the conditions on arbitrary factors α and β_{(k)} which would guarantee the dimensions defined via the above trace to be an a-m functional.

Let us note that although the dimension defined by the usual trace is an a-m functional such a trace is not a morphism of the category if the braiding differs from the flip^{12}. This is the reason why we should modify the usual trace by means of the matrix C (and all its extensions C_{λ}) multiplied by a proper factor α (3.5).

The fact that the matrix C extends to End(V^{⊗m}) in a ”group like” way

C → C^{⊗m}

ensures the conservation of the additive and multiplicative structure of the functional (3.9) restricted on the identity operator (what is just categorical dimension) iff we put

β_{(k)} = α^{k}.  (3.12)

So, if we want the trace tr_{V_{λ}} to be a morphism and giving rise to an a-m functional we have the only free parameter α at our disposal. This parameter (a normalization of the trace) can be found from the following condition dim_{q} V_{(1^{p})} = 1 (see footnote 10). The following lemma plays the key role in finding such a normalization.

Lemma 18 For any Hecke symmetry of the rank p the following relation holds:

Tr_{(1...k)}(A^{(k)}C_{1} … C_{k}) = q^{−pk} \frac{p_{q}!}{k_{q}!(p − k)_{q}!}, 1 ≤ k ≤ p  (3.13)

where the matrix A^{(k)} is the image of k-th order antisymmetrizer A^{(k)} under the representation ρ_{R} (2.1).

Proof. For an arbitrary m ∈ ℤ introduce the auxiliary notation

R_{i}(m) := R_{i} − \frac{q^{m}}{m_{q}}

where m_{q} is defined in (1.9). With this notation the matrix A^{(m)}(R_{1}, …, R_{m−1}) can be presented in the form:

A^{(m)} = \frac{(−1)}{m_{q}}^{m−1}A^{(m−1)}R_{m−1}(m − 1)R_{m−2}(m − 2) … R_{1}(1), A^{(1)} := id.

12For example, if our category SW(V) is supplied with the comodule structure over the RT T Hopf algebra (2.14) then the usual trace is not compatible with the comodule structure.

For the sake of shortness we introduce the notation:

Tr_{q (1...k)} X := Tr_{(1...k)} (XC_{1} … C_{k}).

Now we prove (3.13) by induction. For k = 1 we get with the help of (3.7)

Tr C = q^{−p}p_{q}

which gives the base of induction. Suppose that the assertion of the lemma is valid up to some integer k < p. Then using the simple relations

Tr_{q (k+1)}R_{k}(k) = −q^{−p} \frac{(p − k)_{q}}{k_{q}} id_{(k)}

A^{(k)}R_{k−1}(k − 1) … R_{1}(1) = (−1)^{k−1}k_{q} A^{(k)}.

one can complete the induction

Tr_{q (1...k+1)} A^{(k+1)} = \frac{(−1)^{k}}{(k + 1)_{q}} Tr_{q (1...k+1)} (A^{(k)}R_{k}(k) … R_{1}(1))
= \frac{(−1)^{k}}{(k + 1)_{q}} Tr_{q (1...k)} (A^{(k)} [Tr_{q (k+1)}R_{k}(k)] R_{k−1}(k − 1) … R_{1}(1))
= (−1)^{k+1} q^{−p} \frac{(p − k)_{q}}{k_{q}(k + 1)_{q}} Tr_{q (1...k)} (A^{(k)}R_{k−1}(k − 1) … R_{1}(1))
= q^{−p} \frac{(p − k)_{q}}{(k + 1)_{q}} Tr_{q (1...k)}A^{(k)} = \frac{q^{−p(k+1)} p_{q}!}{(k + 1)_{q}!(p − k − 1)_{q}!}.

This lemma implies that if we take α = q^{p} (which is equivalent to simple renormalization C → q^{p}C in all formulas for the trace) then we get dim_{q}(V_{(1^{p})} = 1 and the categorical dimension related to the trace becomes an a-m functional well defined on objects of our category and hence on the corresponding Grothendieck ring.

Let us summarize the above consideration in the following proposition.

Proposition 19 For any basic object V_{λ}, λ ⊢ m, of the category SW(V) define the trace tr_{V_{λ}} as a morphism End(V_{λ}) → 𝕂 of the form:

∀ F ∈ End(V_{λ}) : tr_{V_{λ}}(F) = q^{pm} Tr_{(1...m)}(F · C_{λ}).

Then such a trace is a morphism in the category SW(V) and the corresponding categorical dimension is an a-m functional. The dimension of any basic object V_{λ} is as follows:

dim_{q}V_{λ} := tr_{V_{λ}}(id) = s_{λ}(q^{p−1}, q^{p−3}, …, q^{1−p})  (3.14)

s_{λ} being the Schur symmetric function (polynomial) in p variables.

Proof. To prove (3.14) we use Proposition 16 and Lemma 18. By virtue of the trace definition one gets from (3.13):

dim_{q}V_{(1^{k})} = q^{pk} Tr_{q (1...k)}A^{(k)} = \frac{p_{q}!}{k_{q}!(p − k)_{q}!} ≡ f_{k}.

Therefore due to Proposition 16 we should find the roots of the polynomial ϕ(t) (3.2) with the above f_{k}. But as is well known the generating function for such f_{k} (which are q-binomial coefficients) is as follows

E_{q}(t) := ∏_{k=0}^{p−1}(q^{2k+1−p} + t), f_{p−k} = [\frac{d^{k}}{dt^{k}} E_{q}(t)]_{t=0} ⇒ ϕ(t) ≡ E_{q}(t).

Therefore the roots of the polynomial ϕ(t) with the coefficients f_{k} = dim_{q} V_{(1^{k})} are the numbers (−q^{2k+1−p}), k = 0, 1, …, p − 1 and result (3.14) is now follows from Proposition 16.

Remark 20 We want to complete the paper with the following observation concerning the Koszul complexes considered in [G]. As was shown there, the Poincaré series P_{±}(t)

P_{±}(t) := ∑_{l} dim Λ_{±}^{l}t^{l}

of ”symmetric” and ”skewsymmetric” algebras Λ_{±}(V) (see (1.10)) satisfy the relation

P_{+}(t) P (−t) = 1.  (3.15)

We would like to point out that if we replace the usual dimensions in formula (3.15) by categorical ones relation (3.15) will be still valid. Moreover, the same is true if we replace the dimensions by the values of any a-m functional. It is not surprising since this fact reflects the well known relation between elementary and complete symmetric functions. And any a-m functional is nothing but a specialization of symmetric functions.

Thus, without constructing any deformed Koszul complex we can obtain some numerical characteristics of quantum objects merely replacing the usual dimensions by their q-analogs (if we disregard the property of the quantum differential to be a morphism of the category).

References

[BW] J.W.Barrett and B.W.Westbery, Spherical Categories, Advances in Mathematics 143 (1999), pp 357-375.

[CP] V.Chari and A. Pressley, A guide to Quantum Groups, Cambrige University Press, 1994.

[DM] P.Deligne and J.S.Miln, Tannakian Categories in: P.Deligne, J.S.Miln, A.Ogus, Shing Kuang-yen Hodges Cycles, Motives, and Shimura Varieties, Lect. Notes in Math. 900, Berlin, Heidelberg, NY: Springer-Verlag, 1982, pp. 101-228.

[DJ] R. Dipper and G.James, Representations of Hecke algebras of General Linear Groups. Proc. London Math. Soc., 52(3) (1986) 20-52. R. Dipper and G.James, Block and Idempotents of Hecke algebras of General Linear Groups, Proc. London Math. Soc., 54(3) (1987) 57-82.

[DP] A.Dold and D.Puppe, Duality, trace and transfer (Russian), Topology (Moscow 1979), Trudy Mat. Ins. Steklov 154 (154), pp 81-97.

[FRT] L.D.Faddeev, N.Yu.Reshetikhin and L.A.Takhtajan, Algebra i Analiz 1 (1989) 178 – 206 (in Russian); English translation in: Leningrad Math. J. 1 (1990) 193.

[Go] M.D.Gould, Quantum Groups and Diagonalization of the Braid Generator, Lett. Math. Phys. 24 (1992) pp. 183 – 196.

[G] D.Gurevich, Algebraic aspects of the quantum Yang-Baxter equation, Leningrad Math. J. 2 (1991), pp. 801–828.

[GM] D.Gurevich and Z.Mriss, Schur-Weyl Categories and Non-quasiclassical Weyl Type Formula, Hopf algebras and quantum groups (Brussels, 1998) Lec. Notes. in Pure and Appl. Math. 209, Dekker, NY 2000.

[GPS] D.Gurevich, P.Pyatov and P.Saponov, Hecke Symmetries and Characteristic Relations on Reflection Equation Algebra, Lett. Mat. Phys. 41 (1997) pp. 255-264.

[HIOPT] L.K.Hadjiivanov, A.P.Isaev, O.V.Ogievetsky, P.N.Pyatov and I.T.Todorov, Hecke algebraic properties of dynamical R-matrices. Application to related quantum matrix algebras, q-alg/9712026

[M] I.G.Macdonald, Symmetric functions and Hall Polynomials, Claredon Press, Oxford, 1979.

[McL1] S.MacLane, Natural associativity and commutativity, Rice Univ. Stud. 49 (1963), pp. 28-46.

[McL2] S.MacLane, Categories for the Working Mathematician, Graduate Texts in Mathematics, Springer Verlag, Berlin 1971.

[OP] O.Ogievetsky and P.Pyatov, Lecture on Hecke algebra, based on lectures at the International School ”Symmetries and Integrable systems”, Dubna, 8-11 June, 1999. JINR Publ. Dept., 2000.

[T] V.G.Turaev, Quantum invariants of Knots and 3-Manifolds, Walter de Gruyter, Berlin, NY 1994.