Weyl approach to representation theory of reflection equation algebra

P.A. Saponov^{∗}

Theory Department, Institute for High Energy Physics, 142284 Protvino, Russia

June 14, 2003

Abstract The present paper deals with the representation theory of the reflection equation algebra, connected with a Hecke type R-matrix. Up to some reasonable additional conditions the Rmatrix is arbitrary (not necessary originated from quantum groups). We suggest a universal method of constructing finite dimensional irreducible non-commutative representations in the framework of the Weyl approach well known in the representation theory of classical Lie groups and algebras. With this method a series of irreducible modules is constructed which are parametrized by Young diagrams. The spectrum of central elements s_{k} = T r_{q} L^{k} is calculated in the single-row and single-column representations. A rule for the decomposition of the tensor product of modules into the direct sum of irreducible components is also suggested.

1 Reflection Equation Algebra

Reflection equation and the corresponding algebra which will be called the reflection equation algebra (REA for short) play a significant role in the theory of integrable systems and non-commutative geometry. In application to integrable systems the reflection equation with a spectral parameter is mainly used. First it appears in the work by I. Cherednik [1]. Usually it comprises the information about the behaviour of a system at a boundary, for example, describes the reflection of particles on a boundary of the configuration space.

The reflection equation without a spectral parameter is important for the non-commutative geometry. One of the first applications the corresponding REA found in the theory of differential calculus on quantum groups (see, e.g., [2]). In such a differential calculus REA with the Hecke type R-matrix is a non-commutative analog of the algebra of vector fields on the groups GL(N) or SL(N). Besides, REA serves as a base for a definition of quantum analogs of homogeneous spaces — orbits of the coadjoint representation of a Lie group, as well as quantum analogs of linear bundles over such orbits (see, e.g., [3, 4]).

In this paper we turn to problems of the representation theory of REA without a spectral parameter. We are interested in the following main topics:

i) a construction of finite dimensional non-commutative irreducible representations and the calculation of spectrum (characters) of central elements in these representations;

ii) a rule for the decomposition of the tensor product of irreducible modules into irreducible components.

∗saponov@mx.ihep.su

Before reviewing the known results, we introduce some necessary definitions and notations. Consider an associative algebra L_{q} with the unity e_{L} over the complex field ℂ generated by n^{2}

elements l̂_{i}^{j}, 1 ≤ i, j ≤ n, n being a fixed positive integer. Let the generators satisfy the following quadratic commutation relations

R_{12}L̂_{1}R_{12}L̂_{1} − L̂_{1}R_{12}L̂_{1}R_{12} = 0, L̂_{1} ≡ L̂ ⊗ I,  (1.1)

where the matrix L̂ ∈ Mat_{n}(L_{q}) is composed of l̂_{i}^{j}: L̂ = ∥l̂_{i}^{j}∥. Here the lower index enumerates the rows while the upper one columns. In (1.1) and everywhere below the use is made of the compact matrix notations [5] when the index of an object indicates the vector space to which the object belongs (or in which this object acts). The symbol I stands for the unity matrix whose dimension is always clear from the context of formulae. A numerical n^{2} × n^{2} matrix R is a solution of the Yang-Baxter equation

R_{12}R_{23}R_{12} = R_{23}R_{12}R_{23}.  (1.2)

The algebra L_{q} described above will be called the reflection equation algebra (REA). Impose now several additional conditions on the matrix R. First of them is the Hecke condition

(R − qI)(R + q^{−1}I) = 0.  (1.3)

The parameter q is a fixed nonzero complex number with the only constraint^{1}

q^{k} ≠ 1, ∀ k ∈ ℕ.  (1.4)

As a consequence, the q-analogs of all integers are nonzero

k_{q} ≡ \frac{q^{k} − q^{−k}}{q − q^{−1}} ≠ 0, ∀ k ∈ ℕ.  (1.5)

Besides, we shall suppose the R-matrix to be skew-invertible that is there exists an n^{2} × n^{2} matrix Ψ such that

∑_{a,b} R_{ia}^{jb}Ψ_{bk}^{as} = δ_{i}^{s}δ_{k}^{j} = ∑_{a,b} Ψ_{ia}^{jb}R_{bk}^{as}.

In the compact notations the above formula reads

T r_{(2)}R_{12}Ψ_{23} = P_{13} = T r_{(2)}Ψ_{12}R_{23},  (1.6)

where the symbol T r_{(2)} means the calculation of trace in the second space and P is the permutation matrix.

To formulate the last requirement on R one should consider the connection of the A_{k} series Hecke algebras with the group algebras of finite symmetric groups. One of the simplest definition of the Hecke algebra reads as follows.

Fix a nonzero complex number q. The Hecke algebra of A_{k} series (k ≥ 2) is an associative algebra H_{k}(q) over the complex field ℂ generated by the unit element 1_{H} and k − 1 generators σ_{i} 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 1_{H})(σ_{i} + q^{−1} 1_{H}) = 0 \end{matrix} \} i = 1, 2, … k − 1.

1All our subsequent constructions possess a well defined “classical limit” q → 1. This limit corresponds to REA with an involutive R-matrix: R^{2} = I.

In some cases it proves to be convenient to consider q as a formal parameter and consider the Hecke algebra over the field of rational functions in the indeterminate q. We shall always bear in mind this extension when considering the classical limit q → 1.

Let us treat R as the matrix of a linear operator (in a fixed basis) which acts in the tensor square V^{⊗2} of a finite dimensional vector space V, dim V = n. Then an arbitrary Hecke R-matrix define the local representation of H_{k}(q) in V^{⊗k}

σ_{i} → ρ_{R}(σ_{i}) = R_{ii+1} = I^{⊗(i−1)} ⊗ R ⊗ I^{⊗(k−i−1)} ∈ End(V^{⊗k}).  (1.7)

If the parameter q satisfies (1.4), then for any positive integer k the Hecke algebra H_{k}(q) is known to be isomorphic to the group algebra ℂ[S_{k}] of the k-th order permutation group S_{k}. As a consequence, there exist elements Y_{ν(a)}(σ) ∈ H_{k}(q) which are the q-analogs of the Young idempotents (projectors) widely used in the theory of symmetric groups. These q-idempotents are parametrized by standard Young tableaux ν(a) corresponding to each diagram or equivalently to each partition ν ⊢ k. The number of all standard tableaux ν(a) which one can construct for a given ν will be denoted dim[ν]

dim[ν] = #{ν(a)}.

In the local representation (1.7) the elements Y_{ν}(σ) are realized as some projector operators in V^{⊗k}. With respect to the action of these projectors the space V^{⊗k} is decomposed into the direct sum of subspaces V_{ν} as in the case of the symmetric group

V^{⊗k} = ⨁_{ν⊢k} ⨁_{a=1}^{dim[ν]} V_{ν(a)}, V_{ν(a)} = Y_{ν(a)}(R) ▷ V^{⊗k}.  (1.8)

The projector Y_{ν(a)}(R) = ρ_{R}(Y_{ν(a)}) is given by some polynomial in matrices R_{ii+1}. For detailed treatment of these questions, explicit formulae for q-projectors and the extensive list of original papers the reader is referred to [6].

So, we shall assume that there exists an integer p > 0 such that the image of the q-antisym-metrizer A^{(p+1)}(σ) ∈ H_{k}(q) (∀ k > p) under the local R-matrix representation ρ_{R} is identical zero while the image of the q-antisymmetrizer A^{(p)}(σ) ∈ H_{k}(q) is a unit rank projector in the space V^{⊗k}

∃ p ∈ ℕ : \{ \begin{matrix}A^{(p+1)}(σ) ⟶^{ρ_{R}} A^{(p+1)}(R) ≡ 0, \\ A^{(p)}(σ) ⟶^{ρ_{R}} A^{(p)}(R), rank A^{(p)}(R) = 1. \end{matrix}  (1.9)

Such a number p will be called the symmetry rank of the matrix R. For example, the symmetry rank of R-matrix connected with the quantum universal enveloping algebra U_{q}(sl_{n}) is equal to n. Examples of n^{2} × n^{2} R-matrices with p < n (for n ≥ 3) were found in [7].

Introduce now two n × n matrices B and C

B_{1} = T r_{(2)}Ψ_{21}, C_{1} = T r_{(2)}Ψ_{12},  (1.10)

where Ψ is defined in (1.6). If the R-matrix has the symmetry rank p these matrices are nonsingular and their product is a multiple of the unit matrix [7]

B · C = \frac{1}{q^{2p}} I.  (1.11)

Besides, B and C have the following traces

T rB = T rC = \frac{p_{q}}{q^{p}} .  (1.12)

The matrices B and C play the central role in what follows.

The simplest example of REA is obtained by choosing the U_{q}(sl_{2}) R-matrix (n = 2)

R = ( \begin{matrix}q & 0 & 0 & 0 \\ 0 & λ & 1 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & q \end{matrix} ) λ ≡ q − q^{−1},  L̂ = ( \begin{matrix}â & b̂ \\ ĉ & d̂ \end{matrix}) .

In this case equation (1.1) leads to six permutation relations for the generators of REA

\begin{matrix}q^{2}âb̂ = b̂â & q(b̂ĉ − ĉb̂) = λ â(d̂− â) \\ q^{2}ĉâ = âĉ & q(ĉd̂− d̂ĉ) = λ ĉâ \\ âd̂= d̂â & q(d̂b̂ − b̂d̂) = λ âb̂. \end{matrix}  (1.13)

Consider a map T r_{q} : Mat_{n}(L_{q}) → L_{q} which is called the quantum trace [5]

T r_{q}(X) =^{def} T r(C · X), X ∈ Mat_{n}(L_{q}).  (1.14)

One can show that the quantities

s_{m}(L̂) = T r_{q}( L̂^{m}) 1 ≤ m ≤ p − 1  (1.15)

are independent central elements of REA (see [5]). Presumably these elements (together with e_{L}) generate the whole center of REA, but we do not know the proof of this hypothesis. Calculation of spectrum of central elements s_{m} in irreducible representations of L_{q} is one of our aims.

At present there exist rather lot of works, devoted to the representation theory of REA. First of all, this algebra possesses a large number of one-dimensional (commutative) representations. For example, it is evident that at any choice of R-matrix relation (1.1) will be satisfied if one sets l̂_{i}^{j} = α δ_{i}^{j}. Less trivial representation can be obtained for our simple example (1.13) by putting â = 0. Then the remaining generators are represented by three arbitrary complex numbers. Such like representations were considered in detail in [8] for the REA with R-matrices coming from U_{q}(sl_{n}) and its super-symmetric generalizations.

But since REA is a non-commutative algebra, the images of a part of its generators are inevitably zero in any one-dimensional representation. That is the kernel of any one-dimensional representation of REA must contain some of its generators. These representations are not comprised by our approach and we shall not consider them.

The main object of our interest will be the non-commutative representations which for any generator of REA put into correspondence a nontrivial linear operator in a finite dimensional vector space. An example of such a representation can be constructed in the following way. Let us use the fact that REA (1.1) is an adjoint comodule over some Hopf algebra which is similar to to the algebra of functions over the quantum group. Suppose the commutations among the generators t_{i}^{j} of the Hopf algebra to be given by the matrix relation [5]

R_{12}T_{1}T_{2} = T_{1}T_{2}R_{12}.  (1.16)

The above multiplication is compatible with the comultiplication Δ

t_{i}^{j} ⟶^{Δ} ∑_{k} t_{i}^{k} ⊗ t_{k}^{j}.  (1.17)

Then (1.1) is covariant with respect to the transformation [9]

l̂_{i}^{j} → t_{i}^{k}S(t_{p}^{j}) ⊗ l̂_{k}^{p},  (1.18)

where S(·) stands for the antipodal map^{2}. So, if one knows the representations of Hopf algebra (1.16) then, given a representation of REA, one can construct another one on the base of (1.18).

Moreover, in the case of U_{q}(sl_{n}) R-matrix one has an additional possibility of constructing non-commutative representations. The matter is that in such a case there exists an embedding of REA into U_{q}(gl_{n}). At the level of generators this embedding is described by the formula [5]

L̂ = S(L^{−})L^{+},  (1.19)

where L^{±} are matrices composed of the U_{q}(gl_{n}) generators. Therefore, starting from a representation of the quantum group one can find the corresponding representation of REA by means of (1.19). In recent paper [10] this approach was extended to the case of an arbitrary quasitriangular Hopf algebra. The authors of the cited paper constructed a universal solution of (1.1) basing on the universal R-matrix of a quasitriangular Hopf algebra H. The generators l̂_{i}^{j} turn out to be elements of the tensor product of H and its “twisted dual” algebra.

However, it is worth pointing out that all the methods mentioned above are essentially based on the representation theory of objects which are external to REA, namely, on the theory of quasitriangular Hopf algebras, the quantum groups being a particular case of them. But as was shown in [11], the Yang-Baxter equation possesses a lot of solutions do not connected with a quantum group (see, also, [7]). For such type solutions we cannot use map (1.19) and cannot construct the REA representations on the base of quantum group ones.

Moreover, having fixed the R-matrix in (1.1), one completely defines all properties of REA and its representation theory as well. The situation is similar to the Lie algebra theory, where the set of structure constants defines all properties of the algebra. Therefore, it is quite natural to develop the representation theory of REA using only the given R-matrix, that is entirely in terms of REA itself.

The most efficient method to solve this problem seems to consist in the direct analysis of the explicit commutation relations of the REA generators. For U_{q}(sl_{n}) R-matrix (at small values of n) one can proceed in an analogy with the representation theory of the universal enveloping algebra of a (simple) Lie algebra. Following this way, P.P. Kulish [12] succeeded in finding all highest vector representations of the simplest REA (1.13). Besides two one-dimensional representations this algebra has a series of finite dimensional irreducible non-commutative representations and one infinite dimensional representation — an analog of the Verma module of the universal enveloping algebra.

Unfortunately, this approach is not universal. It is in essential dependence on the particular choice of R-matrix. The explicit components of matrix relations (1.1) may become completely different when we change the R-matrix. Therefore, one would have to repeat the analysis of commutation relations from the very beginning for each possible R-matrix. Another obstacle in this way is more technical. The matter is that even in the case of U_{q}(sl_{n}) R-matrix the complexity of the explicit form of (1.1) increases very quickly with growing of n. This leads to additional difficulties as compared with the case of universal enveloping algebra, when the commutation relations among generators can be written in a compact form for an arbitrary n.

2Note, that it is skew-invertibility (1.6) which allows one to define the antipode in (bi)algebra (1.16) (to be more precise, in some its extension, see [5]).

In the present paper we suggest a universal method of constructing finite dimensional representations of REA generated by (1.1). These representations are parametrized by Young diagrams and exist for any R-matrix satisfying the additional conditions (1.3), (1.6) and (1.9). For the representations corresponding to single-row and single-column diagrams we calculate the spectrum of central elements (1.15). In the particular case of U_{q}(sl_{2}) R-matrix our result reproduces the series of finite dimensional representations of REA (1.13) obtained in [12].

The paper is organized as follows. In Section 2 we construct an irreducible representation of REA with an arbitrary Hecke R-matrix possessing a finite symmetry rank. This representation is called the fundamental one (of B type) since its tensor products are decomposed into irreducible components similarly to those of fundamental vector representation of U(gl_{n}).

In Section 3 we study the k-th tensor power of the fundamental module of B type and consider its decomposition into higher dimensional REA modules.

Section 4 is devoted to another fundamental module (of R type). The construction is based on the general theory of dual Hopf algebras and can be easily generalized to the case of an arbitrary Hecke R-matrix. The connection of B and R type fundamental modules is established. We also consider an example of reducible indecomposable module over REA which is not equivalent to either B or R type module. At the end of the section we give a short resumé of the obtained results and mention some open questions of the suggested approach.

2 Fundamental module of B type

Consider the REA generated by relations (1.1) and make the linear shift of generators

l_{i}^{j} = l̂_{i}^{j} + \frac{1}{λ}δ_{i}^{j} e_{L},  (2.1)

where e_{L} is the unit element of L_{q} and λ = q − q^{−1}. On taking into account the Hecke condition (1.3) one obtains the commutation relations for the new generators

R_{12}L_{1}R_{12}L_{1} − L_{1}R_{12}L_{1}R_{12} = R_{12}L_{1} − L_{1}R_{12}.  (2.2)

In what follows we shall call the algebra generated by (2.2) the modified reflection equation algebra (mREA) and retain the notation L_{q} for it. Note, that unless q = 1 mREA is isomorphic to REA with relations (1.1). As a consequence, any representation of mREA can be transformed into that of REA and vice versa. Nevertheless, these algebras are different at the classical limit since isomorphism (2.1) is broken at q → 1 in virtue of singularity of λ^{−1}.

In the particular case of U_{q}(sl_{n}) R-matrix the classical limit of (2.2) gives the commutation relations of the U(gl_{n}) generators.

So, consider the mREA L_{q} generated by the unit element and n^{2} generators l_{i}^{j} with commutation relations (2.2). Let us take an n-dimensional vector space V and fix an arbitrary basis of n vectors e_{i}, 1 ≤ i ≤ n. Define a linear map π : L_{q} → End(V) in accordance with the rules

\begin{matrix}π(e_{L}) = id_{V} \\ π(l_{i}^{j}) ▷ e_{k} = e_{i}B_{k}^{j}, \\ π(l_{1} · l_{2} · . . . · l_{k}) = π(l_{1}) · π(l_{2}) · . . . · π(l_{k}), ∀ k ∈ ℕ, \end{matrix}  (2.3)

where the matrix B is defined in (1.10) and id_{V} is the identical operator on V.

Proposition 1 ([15]) The linear map (2.3) defines an irreducible representation of L_{q} (2.2) in the space V. This will be called the fundamental module of B type.

Proof To prove that π realizes the representation of L_{q} one should only verify that operators (2.3) do satisfy (2.2). This can be easily done by a straightforward calculation. The only fact needed in this way consists in the following simple consequence of (1.6) and (1.10)

T r_{(1)}B_{1}R_{12} = I.  (2.4)

Irreducibility follows from the non-singularity of B (1.11). Using this fact one can show that the operators π(l_{i}^{j}) span End(V) and, therefore, the space V does not contain proper invariant subspaces with respect to π.

If we make shift (2.1) in our example (1.13) we get the commutation relations of the corresponding mREA

\begin{matrix}q^{2}ab − ba = qb & q(bc − cb) = (λa − 1)(d − a) \\ q^{2}ca − ac = qc & q(cd − dc) = c(λa − 1) \\ ad = da & q(db − bd) = (λa − 1)b. \end{matrix}  (2.5)

Here in the right column the unit stands for e_{L}. The matrices B and C have the form

B = ( \begin{matrix}q^{−1} & 0 \\ 0 & q^{−3} \end{matrix} ) , C = ( \begin{matrix}q^{−3} & 0 \\ 0 & q^{−1} \end{matrix} ) .

The fundamental representation (2.3) reads

π(a) = ( \begin{matrix}q^{−1} & 0 \\ 0 & 0 \end{matrix} ) , π(b) = ( \begin{matrix}0 & q^{−3} \\ 0 & 0 \end{matrix} ) , π(c) = ( \begin{matrix}0 & 0 \\ q^{−1} & 0 \end{matrix} ) , π(d) = ( \begin{matrix}0 & 0 \\ 0 & q^{−3} \end{matrix} ) .  (2.6)

Given a representation of L_{q} one can find the corresponding representation of the quotient algebra

SL_{q} = L_{q}/{T r_{q}L},  (2.7)

where {X} stands for the ideal generated by a given subset X ⊂ L_{q}. The commutation relations among the generators f_{i}^{j} of SL_{q} has the same form (2.2) as those of L_{q} (with substitution L → F where F = ∥f_{i}^{j}∥), but now the generators are linear dependent due to T r_{q}F = 0.

At q → 1 in the case of U_{q}(sl_{n}) R-matrix the commutation relations of the SL_{q} generators transform into those of U(sl_{n}) generators. For this reason the passage from L_{q} to SL_{q} (or from the L_{q} representation to the corresponding SL_{q} one) will be loosely called the sl-reduction in what follows.

The transformation of an irreducible L_{q} representation ρ acting in a finite dimensional space V into the SL_{q} representation ρ̄ is realized as follows. Due to T r_{q}L is a central element of L_{q} and ρ is an irreducible representation one gets

ρ(T r_{q}L) = χ(T r_{q}L) id_{V} ≡ χ_{1} id_{V} ,

where χ : Z(L_{q}) → ℂ is a character of the center Z(L_{q}). Then the straightforward calculation shows that the SL_{q} generators f_{i}^{j} in representation ρ̄ are given by

ρ̄(f_{i}^{j}) = \frac{1}{ω} (ρ(l_{i}^{j}) − δ_{i}^{j} \frac{χ_{1}}{T rC} id_{V} ), ω = 1 − λ \frac{χ_{1}}{T rC} .  (2.8)

The traceless property ρ̄(T r_{q}F) = 0 is evident and the factor ω^{−1} ensures the correct normalization of the right hand side of (2.2).

Remark 1 As can be easily seen from definition (1.1) the REA admits the “renormalization” automorphism l̂_{i}^{j} → zl̂_{i}^{j} with nonzero complex number z. The same is true for the REA representations as well. At the level of mREA representations this automorphism reads

ρ(l_{i}^{j}) → ρ_{z}(l_{i}^{j}) = zρ(l_{i}^{j}) + δ_{i}^{j} \frac{1 − z}{λ} id_{V} ,  (2.9)

where ρ is an arbitrary mREA representation in the space V. Basing on (2.8) one can show that the corresponding SL_{q} representation ρ̄ does not depend on z that is the whole class of mREA representations ρ_{z} connected by renormalization automorphism (2.9) gives the same SL_{q} representation ρ̄.

Let us now obtain the sl-reduction of the B type representation π defined by (2.3). In virtue of (1.11) one finds

χ_{1} = χ(T r_{q}L) = q^{−2p}.

Then, taking into account (1.12) and (2.8) we find the B type representation π̄ of algebra (2.7)

π̄(f_{i}^{j}) = \frac{1}{ω} (π(l_{i}^{j}) − \frac{δ_{i}^{j}}{q^{p}p_{q}} id_{V} ), ω = \frac{q^{1−p}}{p_{q}} (q^{p−2}(p + 1)_{q} − 1).  (2.10)

Consider again our example (2.5) of mREA L_{q}. To get the corresponding algebra SL_{q} it is necessary to take the quotient of L_{q} over the ideal generated by T r_{q}L. Using the explicit form of C one has

T r_{q}L = \frac{1}{q^{3}} a + \frac{1}{q} d.

With a new generator h = a − d one can rewrite the commutation relations for SL_{q} in terms of three independent quantities b, c and h

\begin{matrix}q^{2}hb − bh = 2_{q}b \\ hc − q^{2}ch = −2_{q}c \\ q(bc − cb) = h(1 − \frac{qλ}{2_{q}} h). \end{matrix}  (2.11)

Note, that at q → 1 relations (2.11) transforms into the well known commutation relations for the generators of the universal enveloping algebra U(sl_{2}).

Starting from (2.6) we have due to (2.10)

π̄(h) = ξ ( \begin{matrix}q & 0 \\ 0 & −q^{−1} \end{matrix} ) , π̄(b) = ξ ( \begin{matrix}0 & q^{−1} \\ 0 & 0 \end{matrix} ) , π̄(c) = ξ ( \begin{matrix}0 & 0 \\ q & 0 \end{matrix} ) , ξ = \frac{q^{2} + 1}{q^{4} + 1}.

At the classical limit this representation turns into the fundamental vector representation of U(sl_{2}).

Let us point out that at q ≠ ±1 the usual trace of π̄(h) does not equal to zero (this property is restored at the classical limit only). However, the quantum trace of the matrix π̄(h) is equal to zero

T r(C · π̄(h)) = 0.

It should be emphasized that in the above relation the matrix C is used to deform the trace of operators of SL_{q} representation, while in (1.14) the quantum trace is taken in Mat_{n}(SL_{q}) (or in Mat_{n}(L_{q})).

So, in order to retain the traceless property of SL_{q} representation in the space V one has to modify the definition of the operator trace in End(V): T r → T r_{q}. This fact is no mere chance. The usual tensor category like that of modules over U(sl_{n}) is not suitable as the representation category for the algebra L_{q} (or SL_{q}). The natural representation category for the mentioned algebras is some quasitensor^{3} Schur-Weyl category. It is the quantum trace (contrary to the usual one) which turns out to be a natural morphism closely connected with the structure of the Schur-Weyl category. The detailed description of the category, the role of the quantum trace in it and the connection with REA are considered in [14] and [15].

To complete the section, we clarify the connection of B type representation (2.6) of mREA (2.5) with the result of [12]. Applying the inverse linear shift of generators (see (2.1)) to representation (2.6) one gets the representation of REA with quadratic relations (1.13). The matrices thus obtained are equivalent (connected by a similarity transform) to the matrices of the twodimensional representation derived in [12]. This representation is the lowest one in the series of non-commutative finite dimensional representations of (1.13). The higher dimensional modules of this series are equivalent to the q-symmetrical tensor powers of the B type modules. Their construction is considered in the next section. So, in the particular case of U_{q}(sl_{2}) R-matrix our approach gives the known result of [12].

3 Higher dimensional modules of B type

Let us turn to the problem of tensor product of mREA modules. We are mainly interested in the following questions. Firstly, given the fundamental module V of B type, how to define an mREA module structure in the tensor power V^{⊗k}? Secondly, into which irreducible higher dimensional modules one can decompose the module V^{⊗k}? At last, how one can decompose the tensor product of arbitrary higher dimensional modules (not only fundamental ones) into the sum of irreducible components? Here we propose answers to these questions.

3.1 Tensor product of fundamental modules

When trying to define the mREA module structure on the tensor product of fundamental modules one finds a serious difficulty. The matter is that it is not known if mREA (2.2) (as well as REA (1.1)) possesses the bialgebra structure. As a consequence, in algebra (2.2) one cannot define the coproduct operation.

To clarify the importance of this operation, consider the case of the universal enveloping algebra U = U(sl_{n}) and dwell upon the definition of the U-module structure on the tensor product of fundamental modules in the framework of Weyl approach [16].

The algebra U is a bialgebra^{4} with the cocommutative coproduct

Δ : U → U ⊗ U.

The action of Δ on a (Lie) generator^{5} x of U is given by a simple formula

Δ(x) = x ⊗ 1_{U} + 1_{U} ⊗ x =^{def} x_{(1)} + x_{(2)},

3The notion of the quasitensor category as a category of finite dimensional representations of a quasitriangular Hopf algebra was introduced in [13].

4Moreover, the algebra U (as the universal enveloping algebra of any Lie algebra) is a Hopf algebra. But for our present purposes only the coproduct is needed.

5As is known, a Lie algebra can be always embedded into its universal enveloping algebra. Here x is the image of a Lie algebra generator under such an embedding.

where 1_{U} is the unit element of U.

Take now some irreducible representation ρ : U → End(V) of U in a finite dimensional vector space V. To get a representation of U in V^{⊗k} one first construct a homomorphism Δ^{k} : U → U^{⊗k} by multiple application of the coproduct Δ. Then the image of U under such a homomorphism is represented in V^{⊗k} with the help of the map ρ^{⊗k}. For a given generator x of U these two steps can be written in the explicit form

i) x ⟶^{Δ^{k}} x = x_{(1)} + x_{(2)} + … + x_{(k)} ∈ U^{⊗k}
ii) x → ρ^{⊗k}(x) ∈ End(V^{⊗k}).  (3.1)

This representation is reducible. To extract the irreducible components one uses the fact that in V^{⊗k} it is possible to define a natural representation of the group algebra ℂ[S_{k}] of the k-th order permutation group S_{k}. With respect to this representation the space V^{⊗k} is decomposed into the direct sum of irreducible ℂ[S_{k}] modules V_{ν(a)}. The modules are parametrized by the standard Young tableaux ν(a), corresponding to all possible partitions ν ⊢ k. With respect to representation of U the subspaces V_{ν(a)} are also irreducible. A generator x of U is represented in V_{ν(a)} by the following linear operator

ρ_{ν(a)}(x) = P_{ν(a)} ρ^{⊗k}(x) P_{ν(a)},  (3.2)

where P_{ν(a)} is the Young projector in V^{⊗k} corresponding to the tableau ν(a). The modules parametrized by different tableaux of the same partition ν are equivalent.

Return now to the case of mREA. As was already mentioned, this algebra does not possess the coproduct and for constructing the tensor product of mREA modules we cannot use the above scheme as it stands. But it proves to be possible to generalize the final formulae (3.1) and (3.2) to the case of mREA.

Introduce the useful notation for a chain of R-matrices

R_{i} ≡ R_{ii+1},  R_{(i→j)}^{±1} =^{def} \{ \begin{matrix}R_{i}^{±1}R_{i+1}^{±1} … R_{j}^{±1} & if i < j \\ R_{i}^{±1}R_{i−1}^{±1} … R_{j}^{±1} & if i > j \\ R_{i}^{±1} & if i = j. \end{matrix}  (3.3)

The analog of reducible representation (3.1) is established in the following proposition.

Proposition 2 Consider the fundamental L_{q} module V defined in Proposition 1 and fix a basis e_{i}, 1 ≤ i ≤ n, in V. The tensor product V^{⊗k} is also an L_{q} module. In the basis e_{i_{1}} ⊗ … ⊗ e_{i_{k}} of V^{⊗k} the matrices of operators representing the L_{q} generators are as follows

ρ_{k}^{t}(l_{i}^{j}) = π^{t}(l_{i}^{j}) ⊗ I^{⊗(k−1)} + ∑_{s=1}^{k−1} R_{(s→1)}^{−1}[π^{t}(l_{i}^{j}) ⊗ I^{⊗(k−1)}]R_{(1→s)}^{−1}.  (3.4)

Here ρ_{k}^{t} and π^{t} stand for the transposed matrices.

Proof The proposition is proved by direct calculations. Since the calculations are rather lengthy we shall not reproduce them in full detail, giving instead the list of important intermediate steps with the corresponding results.

We prove the proposition by induction in k. For k = 1 our assertion reduces to Proposition 1 and hence is true. Suppose it be true up to some k − 1 and prove that then it be true for k.

One should verify that operators (3.4) do satisfy the commutation relations (2.2). Let us assign the number k + 1 to the auxiliary space of indices of the L_{q} generators. One has to show

R_{k+1}ρ_{k}(L_{k+1})R_{k+1}ρ_{k}(L_{k+1}) − ρ_{k}(L_{k+1})R_{k+1}ρ_{k}(L_{k+1})R_{k+1} =
R_{k+1}ρ_{k}(L_{k+1}) − ρ_{k}(L_{k+1})R_{k+1}.  (3.5)

It is convenient to make the transposition of the matrices ρ in the above relation and to take into account that in accordance with (2.3) the matrix π(l_{i}^{j}) reads

π^{t}(L_{k+1})_{1} = P_{1 k+1}B_{k+1},

where the unity enumerates the the matrix indices of the representation space. Then, on substituting (3.4) into (3.5) we find that the first summand in the left hand side decomposes into the sum of k^{2} terms, a typical one being as follows

R_{k+1}R_{(n→1)}^{−1}P_{1 k+1}B_{k+1}[Tr_{(1)}R_{(1→n)}^{−1}R_{(m→1)}^{−1}B_{1}R_{1 k+2}]R_{(1→m)}^{−1} ≡ R_{k+1}Q(n, m),

where the last equality is the definition of Q(n, m), 0 ≤ n, m ≤ k − 1. Here in Q(0, m) the unity matrices are substituted for the chains R_{(1→n)}^{−1} and R_{(n→1)}^{−1}. The second summand in the left hand side of (3.5) expands in a similar way but R_{k+1} stands on the right of Q.

In virtue of the supposition of the induction we conclude that it is sufficient to consider just 2k − 1 terms in each summand containing Q(k − 1, n) and Q(n, k − 1). So, one needs to examine the following expression in the left hand side of (3.5)

∑_{n=0}^{k−2}(R_{k+1}(Q(n, k − 1) + Q(k − 1, n)) − (Q(n, k − 1) + Q(k − 1, n))R_{k+1})
+ R_{k+1}Q(k − 1, k − 1) − Q(k − 1, k − 1)R_{k+1}. (3.6)

The proposition will be proved if one could show that this is equal to

R_{k+1}R_{(k−1→1)}^{−1}P_{1 k+1}B_{k+1}R_{(1→k−1)}^{−1} − R_{(k−1→1)}^{−1}P_{1 k+1}B_{k+1}R_{(1→k−1)}^{−1}R_{k+1}.  (3.7)

Consider first the difference, containing Q(n, k − 1), 0 ≤ n ≤ k − 2. Since (see Appendix)

Tr_{(1)}R_{(1→n)}^{−1}R_{(k−1→1)}^{−1}B_{1}R_{1 k+2} = R_{(k−1→2)}^{−1}P_{2 k+2}B_{k+2}R_{(2→n+1)}^{−1} ∀ n ≤ k − 2

and (see, e.g., [17])

R_{12}B_{1}B_{2} = B_{1}B_{2}R_{12}  (3.8)

we find

R_{k+1}(Q(n, k − 1) + Q(k − 1, n)) − (Q(n, k − 1) + Q(k − 1, n))R_{k+1} =
λR_{(n→1)}^{−1}R_{(k−1→1)}^{−1}P_{1 k+1} B_{k+1}P_{1 k+2}B_{k+2}R_{(2→k−1)}^{−1}R_{(1→n)}^{−1}  (3.9)
− λR_{(n→1)}^{−1}R_{(k−1→2)}^{−1}P_{1 k+1}B_{k+1}P_{1 k+2}B_{k+2}R_{(1→k−1)}^{−1}R_{(1→n)}^{−1}

The calculation of the difference with Q(k − 1, k − 1) in (3.6) is more involved. The trace contained in Q(k − 1, k − 1) is as follows (see Appendix)

Tr_{(1)}R_{(1→k−1)}^{−1}R_{(k−1→1)}^{−1}B_{1}R_{1 k+2} = I − λP_{2 k+2}B_{k+2} − λ ∑_{n=2}^{k−1} R_{(n→2)}^{−1}P_{2 k+2}B_{k+2}R_{(2→n)}^{−1}.

At last, it is a matter of straightforward calculation to show that the unit matrix I in the above expression leads to the necessary contribution (3.7) while the other terms exactly cancel the unwanted summands of type (3.9).

3.2 Decomposition of V^{⊗k} into mREA submodules

Our next goal is to find the decomposition of the L_{q} module V^{⊗k} into irreducible submodules that is to find an analog of the classical formula (3.2). This can be done on the base of isomorphism H_{k}(q) ≅ ℂ[S_{k}] which was discussed in the first section. For our construction the most important consequence of such an isomorphism is the existence of q-analogs of primitive Young idempotents Y_{ν}(σ) ∈ H_{k}(q) and decomposition (1.8) of V^{⊗k} into the direct sum of H_{k}(q) modules.

The main result of this section is formulated in the following proposition.

Proposition 3 Consider the mREA L_{q} generated by (2.2) with the Hecke R-matrix possessing the symmetry rank p (see (1.3), (1.6) and (1.9)). Let V be the fundamental L_{q} module of B type with a fixed basis e_{i}, 1 ≤ i ≤ n. According to Proposition 2 the space V^{⊗k} is also an L_{q} module for any positive integer k. Decompose the tensor product V^{⊗k} into the direct sum (1.8).

Then each component V_{ν(a)} of the direct sum is an L_{q} module and the generators l_{i}^{j} are represented by linear operators π̂_{ν(a)}(l_{i}^{j}) ∈ End(V_{ν(a)}) ↪ End(V^{⊗k}). The matrices of these operators in the basis e_{i_{1}} ⊗ … ⊗ e_{i_{k}} of V^{⊗k} are of the form

π_{ν(a)}^{t}(l_{i}^{j}) = Y_{ν(a)}(R) ρ_{k}^{t}(l_{i}^{j}) Y_{ν(a)}(R),  (3.10)

where ρ_{k} is defined in (3.4) of Proposition 2 and the symbol t means the matrix transposition. The modules parametrized by different tableaux of the same partition ν ⊢ k are equivalent.

Proof Consider matrices (3.4) of the L_{q} representation in V^{⊗k}. Let us assign the number k + 1 to the auxiliary space of indices of L_{q} generators and rewrite the commutation relations in form (3.5). The projectors Y_{ν(a)}(R) in (3.10) are some polynomials in matrices R_{i}, 1 ≤ i ≤ k − 1. The proof of the proposition is based on the fact that the matrix ρ_{k}^{t}(L_{k+1}) commute with R_{i} for 1 ≤ i ≤ k − 1. Indeed, taking this for granted for a moment, we conclude that ρ_{k}^{t}(L_{k+1}) commutes with all projectors Y_{ν(a)}(R). Then, due to the orthogonality and the completeness of the set of projectors

Y_{ν(a)}(R) Y_{μ(b)}(R) = δ_{νμ}δ_{ab}Y_{μ(b)}(R),  (3.11)

∑_{ν⊢k} ∑_{a=1}^{dim[ν]} Y_{ν(a)}(R) = id_{V^{⊗k}}

it is easy to see that relation (3.5) allows the projection onto each component V_{ν(a)} of the direct sum (1.8) and the matrices of corresponding representation are given by (3.10).

To prove the commutativity supposed above one needs no particular property of π(l_{i}^{j}) (an arbitrary n × n matrix X can be substituted for π(l_{i}^{j})). The proof consists in a simple calculation and completely based on the Yang-Baxter equation (1.2) and Hecke condition (1.3).

The equivalence of modules parametrized by different Young tableaux of the same partition ν stems from the connection of the corresponding Young idempotents. It can be shown [6] that any two idempotents Y_{ν(a)}(σ) and Y_{ν(b)}(σ) are connected by a similarity transform. Namely, there exists an invertible element F_{ab}(ν|σ) ∈ H_{k}(q) which is a polynomial in σ_{i} such that

F_{ab}(ν|σ) Y_{ν(a)}(σ) F_{ab}^{−1}(ν|σ) = Y_{ν(b)}(σ), 1 ≤ a, b ≤ dim[ν].

In the local representation (1.7) of H_{k}(q) in V^{⊗k} the element F_{ab}(ν|σ) turns into the intertwining operator F_{ab}(R) which ensures the equivalence of π_{ν(a)} and π_{ν(b)}

F_{ab} π_{ν(a)}^{t} F_{ab}^{−1} = F_{ab} Y_{ν(a)} ρ_{k}^{t} Y_{ν(a)} F_{ab}^{−1} = Y_{ν(b)} ρ_{k}^{t} Y_{ν(b)} = π_{ν(b)}^{t}.

The second equality in the above chain of transformations is valid due to commutativity of ρ_{k}^{t} and F_{ab}(R) since the latter is a polynomial in R_{i}, 1 ≤ i ≤ k − 1.

A natural question arises about the irreducibility of modules V_{ν(a)}. Due to some reasons discussed at the end of the paper we suppose the modules to be irreducible, but still we have no general proof of this fact. We state it as a quite plausible hypothesis.

For the representations parametrized by single-row and single-column diagrams the formulae become much simpler and in this case one can explicitly calculate the spectrum of central elements (1.15).

Corollary 3.1 Consider the L_{q}modules V_{ν(a)}, defined in Proposition 3.

i) For partitions ν = (k) and ν = (1^{k}) (for k ≤ p) the matrices of operators representing the L_{q} generators are given by

π_{(k)}^{t}(l_{i}^{j}) = q^{1−k}k_{q} S^{(k)}(R) [π^{t}(l_{i}^{j}) ⊗ I^{⊗(k−1)}] S^{(k)}(R),  (3.12)
π_{[k]}^{t}(l_{i}^{j}) = q^{k−1}k_{q} A^{(k)}(R) [π^{t}(l_{i}^{j}) ⊗ I^{⊗(k−1)}] A^{(k)}(R), k ≤ p,  (3.13)

where S^{(k)} and A^{(k)} are the q-symmetrizer and the q-antisymmetrizer correspondingly.

ii) In the representations π_{(k)} and π_{[k]} the spectrum χ of central elements s_{m} = T r_{q}L^{m} takes the following values

χ_{(k)}(s_{m}) = q^{−m(p+k−1)−p} k_{q}(p + k − 1)_{q}^{m−1},  (3.14)
χ_{[k]}(s_{m}) = q^{−m(p−k+1)−p} k_{q}(p − k + 1)_{q}^{m−1}.  (3.15)

Proof The assertion i) is the direct consequence of (3.10). Indeed, taking into account the relations

\begin{matrix}R_{i}^{±1}S_{12...k}^{(k)} = q^{±1}S_{12...k}^{(k)} = S_{12...k}^{(k)}R_{i}^{±1} \\ R_{i}^{±1}A_{12...k}^{(k)} = −q^{∓1}A_{12...k}^{(k)} = A_{12...k}^{(k)}R_{i}^{±1} \end{matrix} 1 ≤ i ≤ k − 1

and the definition (1.5) of a q-number one immediately gets (3.12) and (3.13) from (3.10) where an arbitrary projector Y_{ν(a)} should be replaced for S^{(k)}(R) or A^{(k)}(R) respectively.

In order to find the values of characters (3.14) and (3.15) we consider π_{(k)}^{t}(T r_{q}L^{m}) (take the q-symmetrical case for the definiteness) and show that this is a multiple of the q-symmetrizer (the identity operator on the subspace V_{(k)}). The factor is equal to the character χ_{(k)}(s_{m}).

Taking into account (1.11) one rewrites the matrix involved in the form

π_{(k)}^{t}(T r_{q}L^{m}) = q^{m(1−k)−2p} k_{q}^{m} S_{12...k}^{(k)}[T r_{(1)}B_{1}S_{12...k}^{(k)}]^{m−1}S_{12...k}^{(k)}.

To calculate the trace T r_{(1)}B_{1}S_{12...k}^{(k)} we use the recurrent relations for the q-(anti)symmetrizers (see, e.g., [7])

S^{(1)}(R) = I  S_{12...k}^{(k)}(R) = \frac{1}{k_{q}} S_{2...k}^{(k−1)}(R)(q^{1−k}I + (k − 1)_{q}R_{12})S_{2...k}^{(k−1)}(R),  (3.16)
A^{(1)}(R) = I  A_{12...k}^{(k)}(R) = \frac{1}{k_{q}} A_{2...k}^{(k−1)}(R)(q^{k−1}I − (k − 1)_{q}R_{12})A_{2...k}^{(k−1)}(R)  (3.17)

where S_{2...k}^{(k−1)} is the q-symmetrizer of the (k−1)-th order acting in components of V^{⊗k} with numbers from 2 till k. Then using (2.4) and (1.12) one finds

T r_{(1)}B_{1}S_{12...k}^{(k)} = q^{−p} \frac{(p + k − 1)_{q}}{k_{q}} S_{2...k}^{(k−1)}.

Having found the above trace one gets

π_{(k)}^{t}(T r_{q}L^{m}) = q^{−m(p+k−1)−p} k_{q}(p + k − 1)_{q}^{m−1} S^{(k)}(R) = χ_{(k)}(s_{m}) id_{V_{(k)}},  (3.18)

which proves (3.14).

It is worth pointing out that in the above formulae the central role belongs to the symmetry rank p of the R-matrix but not to the parameter^{6} n defining the size of R-matrix and the number of generators in the corresponding REA L_{q}. This feature is characteristic of the Shur-Weyl category connected with the considered series of representations π_{ν} [14].

3.3 The sl-reduction

As the next aim, we are going to explore a problem of tensor product of fundamental SL_{q} representations (2.10). Actually, the solution for the problem follows from Proposition 3 and (2.8). The only thing we need is the spectrum of the element T r_{q}L in representations (3.10).

Lemma 1 Let the partition ν ⊢ k be of the height s that is

ν = (ν_{1}, ν_{2}, …, ν_{s}), ∑_{r=1}^{s} ν_{i} = k, ν_{1} ≥ ν_{2} ≥ … ≥ ν_{s} > 0.

Then the spectrum of the central element s_{1} = T r_{q}L in the representation π_{ν(a)} 1 ≤ a ≤ dim[ν] is as follows

χ_{ν}(s_{1}) = q^{−2p} ∑_{r=1}^{s} q^{2r−1−ν_{r}}(ν_{r})_{q},  (3.19)

where p is the symmetry rank of R-matrix and (ν_{r})_{q} is the q-analog of the integer ν_{r} (see definition (1.5)).

Proof Let us find the matrix π_{ν(a)}(T r_{q}L) for an arbitrary ν(a). Using (1.14) and (1.11) one immediately gets from (3.10)

π_{ν(a)}^{t}(T r_{q}L) = q^{−2p} Y_{ν(a)}(R)(J_{1}^{−1} + J_{2}^{−1} + … + J_{k}^{−1})Y_{ν(a)}(R).

Here the matrices J_{k} read

J_{1} = I, J_{i} = R_{(i−1→1)}R_{(1→i−1)} i ≥ 2.  (3.20)

They are the images of the Jucys-Murphy elements J_{i}(σ) of the Hecke algebra H_{k}(q) under the local representation (1.7). The elements J_{i}(σ) generate the maximal commutative subalgebra in H_{k}(q) and have the following important property (see, e.g., [6])

J_{i}(σ)Y_{ν(a)}(σ) = Y_{ν(a)}(σ)J_{i}(σ) = q^{2(c_{i}−r_{i})}Y_{ν(a)}(σ),

6As was already mentioned above there exist examples of R-matrices with p ≠ n [7].

where c_{i} and r_{i} are respectively the coordinates of the column and the row to which the box with the number i belongs. The quantity q^{2(c−r)} is called a content of the (c, r)-th box of a given Young diagram. Here is an example of the diagram ν = (4, 3, 1^{2}) with corresponding contents

1 q^{2} q^{4} q^{6}
q^{−2} 1 q^{2}

(q^{−4}) / (q^{−6})

Therefore we come to the result

π_{ν(a)}^{t}(T r_{q}L) = q^{−2p}(∑_{i=1}^{k} q^{−2(c_{i}−r_{i})})Y_{ν(a)} = χ_{ν}(T r_{q}L)Y_{ν(a)}.  (3.21)

It is obvious that the sum of all contents (or their inverses) of a given tableau ν(a) is completely defined by the partition (diagram) ν and has the same value for all tableaux ν(a) corresponding to a given ν. As a consequence, the character χ_{ν}(T r_{q}L) does not depend on a. Of course, this is in agreement with the equivalence of representations π_{ν(a)} parametrized by different tableaux ν(a) of the same partition ν.

It is a matter of a simple calculation to show that χ_{ν}(T r_{q}L) represented as the sum of all inverse contents of the tableau ν_{(a)} can be written in form (3.19).

It is interesting to point out one peculiarity of the quantum case. Namely, the spectrum of central elements distinguishes the representations more effectively than in the classical case. To show this, take the q as a parameter and extend the complex field ℂ to the field of rational functions in q. The character χ_{ν}(s_{1}) can be identically transformed to the following expression

q^{2p}λ χ_{ν}(s_{1}) = q^{s}s_{q} − ∑_{r=1}^{s} q^{2(r−ν_{r})−1} ≡ q^{s}s_{q} − Ω(ν_{1}, …, ν_{s}).

Given Ω(ν_{1}, …, ν_{s}) as a function in q, one unambiguously restores the values of all ν_{r}, since the numbers 2(r − ν_{r}) form a strictly increasing sequence. Therefore if the height s of a partition ν is fixed and known then the spectrum of the first central element s_{1} = T r_{q}L is sufficient to distinguish the representations. It is not so, of course, for the classical case q → 1.

Now we are in a position to formulate the results for the higher dimensional modules of the algebra SL_{q}.

Proposition 4 Consider the algebra SL_{q} (2.7) with a Hecke R-matrix possessing the symmetry rank p. Let V be the fundamental L_{q} module of B type with a fixed basis e_{i}, 1 ≤ i ≤ n. According to Proposition 2 the tensor product V^{⊗k} is also an L_{q} module for any k ∈ ℕ. The following assertions are true.

i) The space V^{⊗k} is a reducible SL_{q} module. In the basis e_{i_{1}} ⊗…⊗e_{i_{k}} the matrices of operators representing the SL_{q} generators f_{i}^{j} have the following form

ρ̄_{k}^{t}(f_{i}^{j}) = \frac{1}{ω} (ρ_{k}^{t}(l_{i}^{j}) − \frac{δ_{i}^{j}}{p_{q}q^{p}} Z_{k}),  ω = \frac{q^{1−p}}{p_{q}} (q^{p−2}(p + 1)_{q} − 1),  (3.22)

where the symbol t means the matrix transposition, ρ_{k}(l_{i}^{j}) is defined in (3.4) and Z_{k} is given by

Z_{k} = I + ∑_{n=1}^{k−1} R_{(n→1)}^{−1}R_{(1→n)}^{−1}.

ii) Decompose the tensor product V^{⊗k} into the direct sum (1.8). Each component V_{ν(a)} of the direct sum is an SL_{q} submodule in V^{⊗k}. The generators f_{i}^{j} are represented by linear operators with the following matrices

π̄_{ν(a)}^{t}(f_{i}^{j}) = \frac{1}{ω_{ν}} Y_{ν(a)}[ρ_{k}^{t}(l_{i}^{j}) − δ_{i}^{j} \frac{q^{p}}{p_{q}} χ_{ν}(s_{1})I_{12...k}]Y_{ν(a)},  ω_{ν} = 1 − λ \frac{q^{p}}{p_{q}} χ_{ν}(s_{1}), (3.23)

the character χ_{ν}(s_{1}) being defined in (3.19). The modules parametrized by different tableaux ν(a) of the same partition ν ⊢ k are equivalent.

iii) The spectrum χ̄ of the SL_{q} central elements s̄_{m} = T r_{q}F^{m} in representations π̄_{(k)} and π̄_{[k]} corresponding to ν = (k) and ν = (1^{k}) takes the following values

χ̄_{(k)}(s̄_{m}) = q^{−p−m} \frac{k_{q}(p − 1)_{q}(p + k)_{q}}{(p + k − 1)_{q}} \frac{[(p − 1)_{q}^{m−1}(p + k)_{q}^{m−1} + (−1)^{m}k_{q}^{m−1}]}{(q^{p−2}(p + k)_{q} − k_{q})^{m}}  (3.24)

χ̄_{[k]}(s̄_{m}) = q^{−p+m} \frac{k_{q}(p + 1)_{q}(p − k)_{q}}{(p − k + 1)_{q}} \frac{[(p + 1)_{q}^{m−1}(p − k)_{q}^{m−1} + (−1)^{m}k_{q}^{m−1}]}{(q^{p+2}(p − k)_{q} + k_{q})^{m}}  (3.25)

Proof This proposition is the direct consequence of Propositions 2 and 3, Corollary 3.1 and rule (2.8). Indeed, the operators ρ̄_{k}(T r_{q}F) and π̄_{ν(a)}(T r_{q}F) are obviously equal to zero. Basing on Propositions 2 and 3 one can verify that (3.22) and (3.23) do satisfy (2.2) and that the factor ω_{ν}^{−1} ensures the proper normalization of the right hand side of (2.2).

As for the values (3.24) and (3.25), they can be found by straightforward but rather lengthy calculations on the base of (3.23), (3.14) and (3.15).

In the case of U_{q}(sl_{n}) R-matrix we have p = n and at the classical limit q → 1 the spectrum (3.24), (3.25) of the SL_{q} central elements tends to the spectrum of the U(sl_{n}) Casimir elements in the corresponding representations (see, e.g., [18]).

At last, consider the tensor product of two (irreducible) modules V_{μ} and V_{ν} over L_{q} (or SL_{q}) in which the representation operators π_{μ} and π_{ν} are given by (3.10) (or by (3.23)). Using the isomorphism H_{k} ≅ ℂ[S_{k}] one can show (see, e.g., [14]) that the tensor product V_{μ} ⊗V_{ν} is isomorphic to the following direct sum of L_{q} (or SL_{q}) modules V_{σ}

V_{μ} ⊗ V_{ν} ≅ c_{μν}^{σ}V_{σ}.  (3.26)

Here c_{μν}^{σ} are the Littlewood-Richardson coefficients defining a ring structure in the set of Schur symmetric functions.

4 Fundamental module of R type

Besides the fundamental REA module of B type considered in the previous sections, one can construct another module which will be called the fundamental module of R type. In the case of R-matrix connected with U_{q}(sl_{n}) the corresponding representation originates from the general theory of dual Hopf algebras and we generalize it to the case of an arbitrary R-matrix.

4.1 The definition and tensor product decomposition rule

So, suppose at first that R-matrix defining the structure of REA is the Drinfeld-Jimbo R-matrix connected with the quantum universal enveloping algebra U_{q}(sl_{n}). In this case the Hopf algebra (1.16) is an algebra Fun_{q}(GL(n)) of functions on the quantum group [5]. Besides, there exists embedding (1.19) of the corresponding REA L_{q} into U_{q}(gl_{n}) — the dual Hopf algebra to Fun_{q}(GL(n)). As a consequence, it is possible to define a pairing among the generators l̂_{i}^{j} and t_{i}^{j}.

Using the explicit formulae for the paring of U_{q}(gl_{n}) generators L^{±} and Fun_{q}(GL(n)) generators T (see [5]), we get the following result

⟨T_{1}T_{2} … T_{k}, L̂_{k+1}⟩ = R_{(k→1)}R_{(1→k)} ≡ J_{k+1}.  (4.1)

Here we have used the compact notations (3.3) and (3.20) for the chains of R-matrices.

As is known from the Hopf algebra theory, any module over a Hopf algebra H can be transformed into a comodule over its dual Hopf algebra H^{∗} and vice versa. Let us use this fact in order to define a representation of REA L_{q} in a finite dimensional vector space V, dim V = n.

On fixing a basis e_{i}, 1 ≤ i ≤ n, one can convert the space V into a left comodule over the Hopf algebra H = Fun_{q}(GL(n)) by means of the corepresentation δ

δ : V → H ⊗ V, δ(e_{i}) = t_{i}^{j} ⊗ e_{j}

where the summation over the repeated indices is understood. Then in V we get a right L_{q} action by the following rule

e_{1} ◁ L̂_{2} = ⟨T_{1}, L̂_{2}⟩ e_{1} = R_{12}^{2} e_{1}.

This action can be easily expanded to the tensor product V^{⊗k} for any k ∈ ℕ. Indeed the comodule structure of V^{⊗k} is obvious

δ_{k} : V^{⊗k} → H ⊗ V^{⊗k}, e_{1} ⊗ … ⊗ e_{k} ⟶^{δ_{k}} T_{1} … T_{k} ⊗ (e_{1} ⊗ … ⊗ e_{k}).

Hence, taking into account (4.1)

e_{1} ⊗ … ⊗ e_{k} ◁ L̂_{k+1} = J_{k+1} e_{1} ⊗ … ⊗ e_{k}.

It turns out that we can directly generalize the above formulae to the case of an arbitrary R-matrix. The following proposition is easy to verify.

Proposition 5 Consider the REA L_{q} generated by relations (1.1) with an arbitrary R-matrix. The matrix will be treated as that of a linear operator acting in the tensor square of a finite dimensional vector space V, dim V = n. Define a linear map θ_{k} : L_{q} → End(V^{⊗k}) by the following rule

\{ \begin{matrix}θ_{k}(e_{L}) = id_{V^{⊗k}} \\ θ_{k}(L̂_{k+1}) = α J_{k+1} \\ θ_{k}(l̂_{1} · l̂_{2} · . . . · l̂_{m}) = θ_{k}(l̂_{1}) · θ_{k}(l̂_{2}) · . . . · θ_{k}(l̂_{m}), \end{matrix}  (4.2)

where α ≠ 0 is an arbitrary complex number. Then θ_{k} realizes a representation of L_{q} in the space V^{⊗k}.

Proof It is sufficient to substitute the matrices θ_{k}(L̂_{k+1}) = αJ_{k+1} in (1.1) rewritten in the form

R_{k+1}L̂_{k+1}R_{k+1}L̂_{k+1} − L̂_{k+1}R_{k+1}L̂_{k+1}R_{k+1} = 0

and make use of the following consequence of Yang-Baxter equation (1.2)

\begin{matrix}(R_{1} … R_{k})R_{i} = R_{i+1}(R_{1} … R_{k}), 1 ≤ i ≤ k − 1 \\ (R_{k} … R_{1})R_{i} = R_{i−1}(R_{k} … R_{1}), 2 ≤ i ≤ k. \end{matrix}  (4.3)

As for the numeric factor α ≠ 0, it can be arbitrary due to the renormalization automorphism L̂ → αL̂ (see Remark 1).

Note that in proving Proposition 5 we use nothing but the Yang-Baxter equation for the Rmatrix. Therefore, representation (4.2) is valid not only for the quantum group R-matrix but also for an arbitrary solution of the Yang-Baxter equation (even of a non-Hecke type).

At k > 1 the representation θ_{k} is reducible. The Hecke condition (1.3) is needed for extracting the irreducible components of θ_{k}.

Proposition 6 Let REA L_{q} be generated by (1.1) with a Hecke R-matrix. Consider the representation θ_{k} (4.2) in the space V^{⊗k}. Decompose V^{⊗k} into the direct sum of subspaces V_{ν(a)} in accordance with (1.8).

Then each V_{ν(a)} is an L_{q} submodule and the matrices of linear operators representing the generators l̂_{i}^{j} are given by

θ_{ν(a)}(L̂_{k+1}) = Y_{ν(a)}(R) θ_{k}(L̂_{k+1}) Y_{ν(a)}(R).  (4.4)

The modules parametrized by different tableaux of the same partition ν ⊢ k are equivalent.

Proof The proof is based on the fact that q-projectors Y_{ν}(R) are actually polynomials in J_{1}, …, J_{k} for ν ⊢ k (see [6]). Being the images of Jucys-Murphy elements, the operators J_{i} commute^{7} with J_{k+1}. As a consequence, the relation

R_{k+1}θ_{k}(L̂_{k+1})R_{k+1}θ_{k}(L̂_{k+1}) = θ_{k}(L̂_{k+1})R_{k+1}θ_{k}(L̂_{k+1})R_{k+1}

which takes place in End(V^{⊗k}) admits projection into subspaces End(V_{ν}) in accordance with (4.4).

The equivalence of V_{ν(a)} and V_{ν(b)} corresponding to different tableaux of the same partition ν is proved in the same way as in Proposition 3.

It is worth mentioning, that for constructing representations θ_{ν} one has no need the symmetry rank of R to be finite.

As in the case of B type module one can explicitly calculate the spectrum of central elements (1.15) in the representations parametrized by single-row and single-column diagrams.

Corollary 6.1 In the representations parametrized by partitions ν = (k) and ν = (1^{k}) the spectrum χ̂ of the central elements s_{m} = T r_{q}L̂^{m} takes the following values

χ̂_{(k)}(s_{m}) = q^{−p}(q^{−2m}p_{q} + λ \frac{(p + k)_{q}}{(k + 1)_{q}} q^{m(k−1)}[m(k + 1)]_{q})  (4.5)
χ̂_{[k]}(s_{m}) = q^{−p}(q^{2m}p_{q} − λ \frac{(p − k)_{q}}{(k + 1)_{q}} q^{−m(k−1)}[m(k + 1)]_{q})  k ≤ p  (4.6)

7One can verify this fact independently, on the base of (4.3).

Proof We shall consider the case of q-symmetric representation θ_{(k)} for the definiteness. Let us first calculate θ_{(k)}(T r_{q}L̂). In accordance with (4.2) and (4.4) the matrices representing REA generators read

θ_{(k)}(L̂_{k+1}) = S^{(k)}J_{k+1}S^{(k)}.

Taking into account (see [6]) that

S^{(k+1)} = S^{(k)} \frac{J_{k+1} − q^{−2}}{q^{2k} − q^{−2}}  (4.7)

we rewrite the matrix θ_{(k)}(L̂_{k+1}) in the equivalent form

θ_{(k)}(L̂_{k+1}) = λq^{k−1}(k + 1)_{q}S^{(k+1)} + q^{−2}S^{(k)}.

Next, in virtue of

T r_{q}_{(k+1)}S^{(k+1)} = q^{−p} \frac{(p + k)_{q}}{(k + 1)_{q}} S^{(k)}

we come to the final result

θ_{(k)}(T r_{q}L̂) = q^{−p} (q^{−2}p_{q} + λq^{k−1}(p + k)_{q}) S^{(k)} ≡ χ̂_{(k)}(s_{1}) id_{V_{(k)}}.  (4.8)

Then, basing on (4.7) and on the relations

S^{(k+1)}S^{(k)} = S^{(k+1)}, S^{(k+1)}J_{k+1} = q^{2k}S^{(k+1)}

one can prove (4.5) by induction in the power m of T r_{q}L̂^{m}, where (4.8) serves as the first step. The final step of induction gives

χ̂_{(k)}(s_{m}) = q^{−p}(q^{−2m}p_{q} + λq^{m(k−1)}(p + k)_{q} ∑_{r=0}^{m−1} q^{(k+1)(2r+1−m)}).

With substitution t = q^{k+1} one can easily show that

∑_{r=0}^{m−1} q^{(k+1)(2r+1−m)} = \frac{[m(k + 1)]_{q}}{(k + 1)_{q}}

coming thereby to the desired result (4.5).

The representation θ_{1} in the space V itself is irreducible and the matrices of operators representing L_{q} generators are as follows

θ_{1}(L̂_{2}) = R_{12}^{2}.  (4.9)

where the indices of the first space are those of matrices from End(V) and the indices of the second space enumerates the generators of the algebra. The space V with the above L_{q} representation will be called the fundamental module of R type.

The representation of mREA (2.2) obtained from (4.9) by shift (2.1) reads

θ_{1}(L_{2}) = −R_{12},  (4.10)

where we first perform a renormalization of (4.9) by the factor α = −λ^{−1}.

4.2 The sl-reduction

In order to pass from the REA representation θ_{ν(a)} (4.4) to the corresponding representation θ̄_{ν(a)} of the algebra SL_{q} we need to calculate the spectrum of θ_{ν(a)}(T r_{q}L̂).

Lemma 2 Let the partition ν ⊢ k be of the height s that is

ν = (ν_{1}, ν_{2}, …, ν_{s}), ∑_{r=1}^{s} ν_{i} = k, ν_{1} ≥ ν_{2} ≥ … ≥ ν_{s} > 0.

Then the spectrum of the central element s_{1} = T r_{q}L̂ in the representation θ_{ν(a)} 1 ≤ a ≤ dim[ν] is as follows

θ_{ν(a)}(T r_{q}L̂) = ζ_{ν}(s_{1})Y_{ν(a)},  ζ_{ν}(s_{1}) = q^{−p}p_{q} + λ ∑_{r=1}^{s} q^{ν_{r}+1−2r}(ν_{r})_{q},  (4.11)

where p is the symmetry rank of R-matrix and (ν_{r})_{q} is the q-analog of the integer ν_{r} (see definition (1.5)).

Proof The lemma is proved by the straightforward calculation in analogy with the proof of Lemma 1.

Let us consider the representations θ_{(k)} and θ_{[k]} corresponding to single-row and single-column diagrams in more detail. In this case one can explicitly calculate the spectrum of central elements similarly to the B type representation.

Proposition 7 Let the Hecke type R-matrix has the symmetry rank p. Consider the REA representations of R type θ_{(k)} and θ_{[k]} parametrized by partitions ν = (k) and ν = (1^{k}) (4.4). Then the corresponding representations θ̄ of the SL_{q} generators f_{i}^{j} are as follows

θ̄_{(k)}(F_{k+1}) = \frac{q^{1−p}(p + k)_{q}}{q^{2−p}(p + k)_{q} − k_{q}} (S^{(k)}I_{k+1} − \frac{p_{q}(k + 1)_{q}}{(p + k)_{q}} S^{(k+1)})  (4.12)
θ̄_{[k]}(F_{k+1}) = \frac{q^{−1−p}(p − k)_{q}}{q^{−2−p}(p − k)_{q} + k_{q}} (\frac{p_{q}(k + 1)_{q}}{(p − k)_{q}} A^{(k+1)} − A^{(k)}I_{k+1})  (4.13)

The spectrum ζ̄ of the SL_{q} central elements s̄_{m} = T r_{q}F^{m} in these representations takes the following values

ζ̄_{(k)}(s̄_{m}) = q^{−p−m(p−1)} \frac{k_{q}(p − 1)_{q}(p + k)_{q}}{(k + 1)_{q}} \frac{[(p + k)_{q}^{m−1} + (−1)^{m}k_{q}^{m−1}(p − 1)_{q}^{m−1}]}{(q^{2−p}(p + k)_{q} − k_{q})^{m}} (4.14)
ζ̄_{[k]}(s̄_{m}) = q^{−p−m(p+1)} \frac{k_{q}(p + 1)_{q}(p − k)_{q}}{(k + 1)_{q}} \frac{[(−1)^{m}(p − k)_{q}^{m−1} + k_{q}^{m−1}(p + 1)_{q}^{m−1}]}{(q^{−2−p}(p − k)_{q} + k_{q})^{m}} (4.15)

Proof The proof consists in direct calculations on the base of (2.8) and we shall not present it here.

4.3 Interrelation between modules of B and R types

Let us now find a connection between the fundamental modules of B and R types. If the symmetry rank p = 2 (for example, when R stems from U_{q}(sl_{2})) these modules are equivalent. To be more precise, the situation is as follows. In virtue of (1.9) the q-antisymmetrizer A^{(2)} is a unit rank projector in V^{⊗2} and its matrix can be written in the form

A_{i_{1}i_{2}}^{j_{1}j_{2}} = u_{i_{1}i_{2}}v^{j_{1}j_{2}},

the matrices ∥u_{ij}∥ and ∥v^{ij}∥ being nonsingular. Then one can show that the representations π and θ_{1} of mREA (2.2) are connected by the relation

q^{2} u_{1} · π(L_{2}) · u_{1}^{−1} = qI_{12} + θ_{1}(L_{2}).

After the sl-reduction we come to the representations π̄ and θ̄_{1} of the algebra SL_{q} (2.7) and simplify the above formula to the expression

u_{1} · π̄(F_{2}) · u_{1}^{−1} = θ̄_{1}(F_{2}),

F = ∥f_{i}^{j}∥ being the matrix composed of the SL_{q} generators.

In the case p > 2 the fundamental modules of B and R types are not equivalent. Constraining ourselves to the case of SL_{q} algebra (2.7) we shall prove that R type representation θ̄(F) is equivalent to π̄_{[p−1]}(F) obtained from (3.13) by means of sl-reduction (3.23).

For this purpose, consider in more detail the structure of the subspace V_{[p−1]} ⊂ V^{⊗(p−1)}. By definition (1.8) the subspace V_{[p−1]} is the image of the q-antisymmetrizer A^{(p−1)}

V_{[p−1]} = A^{(p−1)}(R) ▷ V^{⊗(p−1)}.

Since the symmetry rank of R-matrix is equal to p then the q-antisymmetrizer A^{(p)} is a unit rank projector and its matrix can be written in the form

A^{(p)}_{i_{1}...i_{p}}^{j_{1}...j_{p}} = u_{i_{1}...i_{p}}v^{j_{1}...j_{p}}  (4.16)

where as follows from (3.11) the tensors u and v are normalized by the condition

∑_{{i}} u_{i_{1}...i_{p}}v^{i_{1}...i_{p}} = 1.

It is convenient to introduce the following linear combinations of the basis vectors of the space V^{⊗(p−1)}

ϵ^{i} =^{def} ∑_{{a}} v^{ia_{2}...a_{p}}e_{a_{2}} ⊗ … ⊗ e_{a_{p}}.  (4.17)

The following lemma establishes an important property of the vectors ϵ^{i}.

Lemma 3 Consider the set of n vectors ϵ^{i} ∈ V^{⊗(p−1)} defined in (4.17). These are eigenvectors of the q-antisymmetrizer A^{(p−1)} and they form a basis of the subspace V_{[p−1]}

A^{(p−1)}(R) ▷ ϵ^{i} = ϵ^{i},  ∀ w ∈ V_{[p−1]} : w = ∑_{i} w_{i}ϵ^{i}.  (4.18)

Proof Consider the recurrence relation (3.17) for the q-antisymmetrizer A^{(p)} and calculate the trace in the first matrix space with the matrix B_{1}

T r_{(1)}B_{1}A_{12...p}^{(p)} = \frac{1}{q^{p}p_{q}} A_{2...p}^{(p−1)}.

In virtue of (4.16) we get the following expression for the matrix 𝔸 of A^{(p−1)}

𝔸_{a_{2}...a_{p}}^{b_{2}...b_{p}} = q^{p}p_{q} ∑_{m,n} B_{m}^{n}u_{na_{2}...a_{p}}v^{mb_{2}...b_{p}},  (4.19)

where for the compactness we omit the superscript (p − 1) of the matrix 𝔸. Also we need the formula connecting the matrix C (1.10) and the tensors u and v. It can be shown (see [7]) that

C_{i}^{j} = \frac{p_{q}}{q^{p}} ∑_{\{a\}} u_{ia_{2}...a_{p}}v^{ja_{2}...a_{p}} ≡ \frac{p_{q}}{q^{p}} ∑_{\{a\}} u_{i\{a\}}v^{j\{a\}},  (4.20)

where in the last equality we have introduced a convenient multi-index notation.

At last, taking into account definition (4.17) we get the necessary result (the summation over the repeated indices is understood)

A^{(p−1)} ▷ ϵ^{i} = v^{ia_{2}...a_{p}}𝔸_{a_{2}...a_{p}}^{b_{2}...b_{p}}e_{b_{2}} ⊗ … ⊗ e_{b_{p}} ≡ v^{i{a}}𝔸_{{a}}^{{b}}e_{{b}}
= q^{p}p_{q} v^{i{a}} e_{{b}} v^{m{b}} B_{m}^{n} u_{n{a}} = q^{p}p_{q} ϵ^{m}B_{m}^{n}u_{n{a}}v^{i{a}}
= q^{2p} ϵ^{m} B_{m}^{n}C_{n}^{i} = ϵ^{i}.

Here at the last step we have used (1.11).

Therefore, under the action of the q-antisymmetrizer A^{(p−1)} the space W = Span{ϵ^{i}} is an invariant subspace in V_{[p−1]} and hence W = V_{[p−1]}. But as was proved in [7]

dim V_{[p−1]} = dim V = n.

Therefore the n vectors ϵ^{i} cannot be linear dependent since otherwise dim V_{[p−1]} = dim W < n. So, the set of eigenvector ϵ^{i} of the q-antisymmetrizer A^{(p−1)} can be taken a basis of V_{[p−1]}.

Now we are ready to establish the connection of B and R type fundamental modules in the case of R-matrix with a finite symmetry rank.

Proposition 8 Let the R-matrix has the symmetry rank p. Then the SL_{q} representation θ̄_{1} obtained from (4.10) is equivalent to π̄_{[p−1]} obtained from (3.13) by sl-reduction (3.23).

Proof Let us first consider the L_{q} representation π_{[p−1]} (3.13) which acts in the subspace V_{[p−1]}. In virtue of Lemma 3 we shall find the matrices of operators π_{[p−1]}(l_{i}^{j}) in the basis of vectors ϵ^{k} (4.17). Using (3.13) we obtain (in the same notations as in the proof of Lemma 3)

\frac{q^{2−p}}{(p − 1)_{q}} π_{[p−1]}(l_{i}^{j}) ▷ ϵ^{k} = v^{k\{a\}}𝔸_{\{a\}}^{m\{c\}}B_{m}^{j}𝔸_{i\{c\}}^{\{b\}}e_{\{b\}} = (use (4.19))
= q^{2p}p_{q}^{2} e_{{b}}v^{r{b}}B_{s}^{l}(v^{k{a}}u_{l{a}})B_{r}^{n}(v^{sm{c}}u_{ni{c}})B_{m}^{j}
= q^{3p}p_{q} ϵ^{r} (B_{s}^{l}C_{l}^{k})B_{r}^{n}(v^{sm{c}}u_{ni{c}})B_{m}^{j} = q^{p}p_{q} ϵ^{r}B_{r}^{n}(v^{km{c}}u_{ni{c}})B_{m}^{j}.

Introduce an n^{2} × n^{2} matrix Ω with matrix elements

Ω_{s_{1}s_{2}}^{r_{1}r_{2}} = p_{q}(p − 1)_{q} ∑_{{a}} u_{s_{1}s_{2}{a}}v^{r_{1}r_{2}{a}}.

Then, the matrix of the operator π_{[p−1]}(l_{i}^{j}) in the basis ϵ^{k} has the form (in compact notations)

(π_{[p−1]}(L_{2}))_{1} = q^{2(p−1)} B_{1}Ω_{12}B_{2}.  (4.21)

With the use of (3.17) for A^{(p)} and (4.20) for C one can express the matrix Ω in a more explicit form. Omitting straightforward calculations we write down the final result

Ω_{12} = q^{2p−1}(C_{1}C_{2} − qC_{1}Ψ_{21}C_{1}),

where Ψ is the skew-inverse to R-matrix as defined in (1.6). Substituting this into (4.21) we find

(π_{[p−1]}(L_{2}))_{1} = q^{−3}I_{12} − q^{2p−2}Ψ_{21}C_{1}B_{2}.

After sl-reduction (3.23) we get the representation of the SL_{q} algebra

(π̄_{[p−1]}(F_{2}))_{1} = \frac{q^{1−p}}{(p − 1)_{q} + q^{p+2}} (I_{12} − q^{3p}p_{q}Ψ_{21}C_{1}B_{2}).

The sl-reduction of the R type representation (4.10) leads in turn to the result

θ̄_{1}(F_{2}) = \frac{q^{p+1}}{(p − 1)_{q} + q^{p+2}} (I_{12} − q^{−p}p_{q} R_{12}).

Next we take into account the connection of Ψ and R (see Appendix)

q^{2p}C_{1}Ψ_{21}B_{2} = R_{12}^{−1}.  (4.22)

With this formula one immediately gets

C_{1} (π̄_{[p−1]}(F_{2}))_{1} C_{1}^{−1} = θ̄_{1}(F_{2})

which means that the corresponding modules are equivalent.

4.4 Indecomposable modules: an example

One can put a natural question about the completeness of the set of representations thus obtained. In other words, whether an arbitrary finite dimensional REA module with a non-commutative representation of (1.1) is equivalent to a direct sum of modules V_{ν} defined in Proposition 3?

The answer to this question is negative. The matter is that REA (1.1) possesses reducible finite dimensional but indecomposable modules. The corresponding mREA representations do not admit a finite classical limit q → 1. Let us give a simplest example of such a module for algebra L_{q} (1.13).

We start from the one-dimensional representation ρ : L_{q} → ℂ (see [8])

ρ(L̂) = ( \begin{matrix}0 & x \\ y & z \end{matrix} ) ,  x, y, z ∈ ℂ.  (4.23)

Then we use the comodule property (1.18) in order to get the higher dimensional representation of L_{q}. For this purpose take the known R-matrix representation γ of (1.16)

γ(T_{1}) = P_{12}R_{12}, γ(S(T_{1})) = R_{12}^{−1}P_{12}.

Here P is the transposition matrix, R is the U_{q}(sl_{2}) R-matrix and the second matrix space stands for the representation space V, dim V = 2. Then in accordance with (1.18) we construct a two dimensional representation ρ_{2} of L_{q} in the space V ⊗ ℂ ≅ V

ρ_{2}(L̂_{1}) = γ(T_{1})ρ(L̂_{1})γ(S(T_{1})) = R_{21}ρ(L̂_{2})R_{21}^{−1}.

For the L_{q} generators (1.13) the explicit form of the representation ρ_{2} reads as follows

ρ_{2}(â) = ( \begin{matrix}0 & −qλx \\ 0 & 0 \end{matrix} ) , ρ_{2}(b̂) = ( \begin{matrix}qx & 0 \\ 0 & q^{−1}x \end{matrix} ) , ρ_{2}(ĉ) = ( \begin{matrix}q^{−1}y & −λz \\ 0 & qy \end{matrix} ) , ρ_{2}(d̂) = ( \begin{matrix}z & q^{−1}λx \\ 0 & z \end{matrix} ) .

The module V with the representation ρ_{2} is reducible. The one-dimensional submodule is spanned by the basis vector e_{1}. The corresponding one-dimensional representation

â → 0, b̂ → qx, ĉ → q^{−1}y, d̂→ z

is connected with the initial one (4.23) by an automorphism η of L_{q} [8]

η ( \begin{matrix}â & b̂ \\ ĉ & d̂ \end{matrix}) = ( \begin{matrix}â & ωb̂ \\ ω^{−1}ĉ & d̂ \end{matrix} ) , ∀ ω ∈ ℂ^{×}.

Nevertheless, being reducible, the module V is obviously indecomposable since matrices ρ_{2}(â) and ρ_{2}(d̂) cannot be transformed into diagonal form (unless x = 0). Therefore, this module cannot be presented as a direct sum of modules V_{ν} constructed in Section 3.

So, examining the completeness of the set of V_{ν} we have to reduce the class of admissible modules to completely reducible ones and reformulate the question in the following way: is any completely reducible finite dimensional module over REA (1.1) isomorphic to a direct sum of modules V_{ν}?

For an arbitrary R-matrix with a finite symmetry rank we have no definite answer to this question. Given the only symmetry rank of R, one has too little information on the concrete structure of the corresponding REA. Perhaps, an analysis of the explicit commutation relations is needed here. The question on irreducibility of modules V_{ν} themselves is also open in this case.

As for the R-matrix originated from the quantum universal enveloping algebra U_{q}(sl_{n}) (p = n) it is highly plausible that the finite direct sums of the modules V_{ν} do exhaust all finite dimensional completely reducible (non-commutative) representations of REA. The matter is that the matrix elements of the corresponding representations are rational functions in q with nonsingular limit q → 1. At that limit the mREA SL_{q} (2.7) tends to the algebra U(sl_{n}) and all the SL_{q} modules V_{ν} go to the corresponding modules over U(sl_{n}). In particular the modules V_{ν} described in Proposition 3 must be irreducible.

To conclude, we shortly summarize the main results and discuss some open problems and perspectives.

For the reflection equation algebra we have constructed the series of finite dimensional noncommutative representations which are parametrized by Young diagrams. The representations exist for any R-matrix satisfying the additional conditions (1.3), (1.6) and (1.9). The corresponding modules V_{ν} are simple objects of a quasitensor Schur-Weyl category described in detail in [14]. As was pointed out in Section 3, the Grothendiek ring of the Schur-Weyl category for the Hecke Rmatrix with the symmetry rank p is isomorphic to that of the category of finite dimensional modules over U(sl_{p}). Nevertheless, dimensions of the modules and the characters of central elements could drastically differ from each other.

Also, it is worth mentioning some further problems in this approach. First of them is the problem of constructing the representation theory for the REA connected with R-matrices of the Birman-Murakami-Wenzl type. Examples are given by R-matrices originated from the quantum groups of B, C and D series. The key point here is to develop the adequate technique for the q-analogs of the Young idempotents.

Another interesting problem is the representation theory for the REA with a spectral parameter. This can find a lot of applications to the theory of integrable systems.

Acknowledgment The author would like to thank D.I. Gurevich and P.N. Pyatov for valuable and helpful discussions on the problems considered in this paper. Also I express my gratitude to the staff and guests of the Max-Planck Institute für Mathematik for kind hospitality, perfect conditions for work and encouraging and friendly atmosphere.

Appendix

This is a technical section where some auxiliary formulae of the main text are proved. First, we prove the trace formulae which were used in Proposition 2. The decisive role belongs to the following result.

Lemma 4 Let R be a solution of the Yang-Baxter equation (1.2), satisfying the additional condition (1.6). Then

T r_{(0)}B_{0}R_{01}R_{02}^{−1} = P_{12}B_{1},  (4.24)

where P is the transposition matrix and B is defined in (1.10).

Proof Rewrite the Yang-Baxter equation (1.2) in the equivalent form

R_{12}R_{23}R_{12}^{−1} = R_{23}^{−1}R_{12}R_{23}.

Using this equation and definition (1.6) of the skew-inverse matrix Ψ we obtain the following relation

T r_{(0)}Ψ_{10}R_{02}R_{03}^{−1} = P_{12}T r_{(0)}R_{10}^{−1}R_{20}Ψ_{03}P_{23}.

Calculate now the trace in the first space. Since T r_{(1)}Ψ_{10} = B_{0}, we get

T r_{(0)}B_{0}R_{02}R_{03}^{−1} = T r_{(01)}R_{20}^{−1}P_{12}R_{20}Ψ_{03}P_{23} = T r_{(0)}Ψ_{03}P_{23} = B_{3}P_{23} = P_{23}B_{2}.

This result differs from (4.24) only in the notations of the matrix spaces.

So, we are ready to prove the trace formulae used in Proposition 2.

i). T (n, k − 1) ≡ T r_{(1)}R_{(1→n)}^{−1}R_{(k−1→1)}^{−1}B_{1}R_{1 k+2} at n < k − 1. First of all, one should use (4.3) in order to draw the chain R_{(1→n)}^{−1} to the right of R_{(k−1→1)}^{−1}

R_{(1→n)}^{−1}R_{(k−1→1)}^{−1} = R_{(k−1→1)}^{−1}R_{(2→n+1)}^{−1}.

The chain R_{(2→n+1)}^{−1} is evidently commute with B_{1}R_{1 k+2}, therefore

T (n, k − 1) = R_{(k−1→2)}^{−1} [T r_{(1)}R_{12}^{−1}B_{1}R_{1 k+2}] R_{(2→n+1)}^{−1}.

Using the cyclic property of trace and then relation (4.24), we come to the desired result

T (n, k − 1) = R_{(k−1→2)}^{−1}P_{2 k+2}B_{k+2}R_{(2→n+1)}^{−1}.

ii) The calculation of T (k − 1, k − 1) is more cumbersome. The main difficulty is that in this case the chains of R-matrices cannot be drawn through each other. As a consequence, it is not so easy to decrease the number of R-matrices with the indices in the first space in order to apply (4.24). However, with the help of the Hecke condition and the Yang-Baxter equation the product of R-matrix chains contained in T (k − 1, k − 1) can be transformed as follows

R_{(1→k−1)}^{−1}R_{(k−1→1)}^{−1} = I_{12...k} − λR_{1}^{−1} − λ ∑_{n=2}^{k−1} R_{(n→2)}^{−1}R_{1}^{−1}R_{(2→n)}^{−1}.

The terms in the right hand side contain at most one R matrix with indices in the first space and hence, upon multiplying by B_{1}R_{1 k+2}, we can calculate T r_{(1)} with the help of (4.24). As a result we get the formula which was used in the proof of Proposition 2

T (k − 1, k − 1) = I_{12...k} − λP_{2 k+2}B_{k+2} − λ ∑_{n=2}^{k−1} R_{(n→2)}^{−1}P_{2 k+2}B_{k+2}R_{(2→n)}^{−1}.

iii) At last, relation (4.22) is a direct consequence of (4.24). Indeed, multiply (4.24) by Ψ_{13} from the right and take the trace in the first space. Due to definition (1.6) of the matrix Ψ we find

T r_{(0)}B_{0}P_{03}R_{02}^{−1} = T r_{(1)}P_{12}B_{1}Ψ_{13} = B_{2}Ψ_{23} T r_{(1)}P_{12} = B_{2}Ψ_{23}.

On the other hand, due to the cyclic property of trace

T r_{(0)}B_{0}P_{03}R_{02}^{−1} = T r_{(0)}P_{03}R_{02}^{−1}B_{0} = R_{32}^{−1}B_{3} T r_{(0)}P_{03} = R_{32}^{−1}B_{3}.

Therefore

B_{2}Ψ_{23} = R_{32}^{−1}B_{3}.

Multiplying this by C_{3} from the right and using (1.11) we come to the relation

q^{2p}B_{2}Ψ_{23}C_{3} = R_{32}^{−1.}

Actually this is equivalent to (4.22), since basing on (3.8) one can easily show that

B_{2}Ψ_{23}C_{3} = C_{3}Ψ_{23}B_{2}.

References

[1] I.V. Cherednik, Teor. Mat. Fiz., vol.61 (1984) 35–44 (in Russian).

[2] P. Schupp, P. Watts, B. Zumino, Lett. Mat. Phys., 25 (1992) 139–147.

B. Zumino, Prep. Univ. of California UCB-PTH-62/91. A.P. Isaev, P. Pyatov, Phys. Lett. A,179 (1993) 81–90.

[3] J. Donin, A. Mudrov, Lett. Mat. Phys. 62 (2002) 17–32.

[4] D. Gurevich, P. Saponov, J. Phys. A, 35 (2002), 9629–9643.

D. Gurevich, P. Saponov, J. Phys. A, 34 (2001), 4553–4569.

[5] N.Yu. Reshetikhin, L.A. Takhtajan, L. D. Faddeev, Alg. i Analiz, vol. 1, (1989), no. 1, 178–206 (in Russian). Engl. transl.: Leningrad Math. J., vol. 1 (1990), no. 1, 193–225.

[6] O. Ogievetsky, P. Pyatov, “Lecture on Hecke algebras”, In “Symmetries and Integrable systems”, Dubna publishing, 2000; Preprint CPT-2000/P.4076.

[7] D.I. Gurevich, Alg. i Analiz, vol. 2, (1990), no.4, 119 (in Russian).

Engl. transl.: Leningrad Math. J., vol. 2 (1991) 801–828.

[8] P.P. Kulish, R. Sasaki, C. Schwiebert, J. Math. Phys. 34 (1993) 286–304.

[9] P. P. Kulish, E. K. Sklyanin, J. Phys. A: Math. Gen., 25 (1992) 5963-5976.

P.P. Kulish, R. Sasaki, Progr. Theor. Phys., 89 (1993) 741–761.

[10] J. Donin, P. P. Kulish, A. I. Mudrov, “On universal solution to reflection equation”,

    math.QA/0210242

[11] A.A. Belavin, V.G. Drinfel’d, Func. Analiz i Pril., vol. 16 (1982), no. 3, 1–29.

[12] P.P. Kulish, Alg. i Analiz, vol. 6 (1994), no. 2, 195–205 (in Russian).

Engl. transl.: St. Petersburg Math. J., vol. 6 (1995), no. 2, 365–374.

[13] N.Yu. Reshetikhin, Alg. i Analiz, vol. 1 (1989), no.2, 169–188 (in Russian).

Engl. transl.: Leningrad Math. J., vol. 1 (1990), no. 2, 491–513

[14] D. Gurevich, R. Leclercq, P. Saponov, J. Geom. Phys., vol. 44 (2002) 251–278.

[15] D. Gurevich, R. Leclercq, P. Saponov, math.QA/0207268.

[16] H. Weyl, “The Classical Groups, their Invariants and Representations”, Princeton Univ. Press, 1973.

[17] D. Gurevich, P. Pyatov, P. Saponov, Lett. Math. Phys., vol. 41 (1997), no. 3, 255–264.

[18] A.O. Barut, R. Ra̧czka, “Theory of Group Representations and Applications”, World Sci. Pub., the Second Ed., 1986.