Abstract. Let R : V ⊗2 → V ⊗2 be a Hecke type solution of the quantum YangBaxter equation (a Hecke symmetry). Then, the Hilbert-Poincaré series of the associated R-exterior algebra of the space V is the ratio of two polynomials of degrees m (numerator) and n (denominator).Under the assumption that R is skew-invertible, a rigid quasitensor category SW(V (m|n) ) of vector spaces is defined, generated by the space V and its dual V * , and certain numerical characteristics of its objects are computed. Moreover, a braided bialgebra structure is introduced in the modified reflection equation algebra associated with R, and the objects of the category SW(V (m|n) ) are equipped with an action of this algebra. In the case related to the quantum group U q (sl(m)), the Poisson counterpart of the modified reflection equation algebra is considered and the semiclassical term of the pairing defined via the categorical (or quantum) trace is computed. §1. IntroductionThe reflection equation algebra is a very useful tool of the theory of integrable systems with boundaries. It derives its name from an equation describing factorized scattering on a half-line (see [C], where the reflection equation depending on a spectral parameter was introduced for the first time).By definition (see [KS]), the reflection equation algebra (REA for short) is an associative unital algebra over a ground field 1 K generated by elements l j i , 1 ≤ i, j ≤ N , subject to the following quadratic commutation relations:Here L 1 = L ⊗ I, L = l j i is the matrix composed of REA generators, while the linear operator R : V ⊗2 → V ⊗2 is an invertible solution of the quantum Yang-Baxter equationHere V is a finite-dimensional vector space over the field K, dim K V = N , and the indices of R correspond to the space (or spaces) in which the operator is applied.
Yangian-like algebras, associated with current R-matrices, different from the Yang ones, are introduced. These algebras are of two types. The so-called braided Yangians are close to the Reflection Equation algebras, arising from involutive or Hecke symmetries. The Yangians of RTT type are close to the corresponding RTT algebras. Some properties of these two classes of the Yangians are studied. Thus, evaluation morphisms for them are constructed, their bi-algebra structures are described, and quantum analogs of certain symmetric polynomials, in particular, quantum determinants, are introduced. It is shown that in any braided Yangian this determinant is always central, whereas in the Yangians of RTT type it is not in general so. Analogs of the Cayley-Hamilton-Newton identity in the braided Yangians are exhibited. A bozonization of the braided Yangians is performed.2. In the Yangian Y(gl(N )) there are well-defined quantum analogs of some symmetric polynomials, namely, the elementary symmetric polynomials and power sums. The highest quantum elementary symmetric polynomial is called the (quantum) determinant. This determinant (more precisely, its Laurent coefficients) generate the center of the Yangian Y(gl(N )). The quantum elementary symmetric polynomials generate a commutative subalgebra in the Yangian Y(gl(N )). The power sums are related with elementary polynomials by a quantum analog of Newton relations and generate the same commutative subalgebra.
On any Reflection Equation algebra corresponding to a skew-invertible Hecke symmetry (i.e. a special type solution of the Quantum Yang-Baxter Equation) we define analogs of the partial derivatives. Together with elements of the initial Reflection Equation algebra they generate a "braided analog" of the Weyl algebra. When q → 1, the braided Weyl algebra corresponding to the Quantum Group U q (sl(2)) goes to the Weyl algebra defined on the algebra Sym((u(2)) or that U (u(2)) depending on the way of passing to the limit. Thus, we define partial derivatives on the algebra U (u(2)), find their "eigenfunctions", and introduce an analog of the Laplace operator on this algebra. Also, we define the "radial part" of this operator, express it in terms of "quantum eigenvalues", and sketch an analog of the de Rham complex on the algebra U (u(2)). Eventual applications of our approach are discussed.Essentially, such a pseudogroup is the famous RTT algebra (see [FRT]) associated with a given braiding, i.e. an invertible operator R : V ⊗2 → V ⊗2 , satisfying the Quantum Yang-Baxter EquationHereafter V is a vector space over the ground field K (= R or C) and I is the identity operator. In [W] a general scheme of defining differential forms and the de Rham complex on a matrix pseudogroup was suggested. Also, the author considered analogs of vector fields, introducing them by duality. In some subsequent publications (see f.e. [IP, FP]) the algebra generated by such fields in the case related to the QG U q (sl(m)) was identified as a (modified) Reflection Equation (RE) algebra 1 (see below).Recently, we have (partially 2 ) generalized this construction replacing the RTT algebra by other quantum matrix algebras, in particular, by an RE algebra. Namely, taking a copy of the RE algebra (denoted M) as a quantum function algebra, we treated another copy of this algebra (denoted L) as an analog of the one-sided or adjoint differential operators. Below, we deal with the total algebra (denoted B(L, M)) where we assume that M is equipped with a left (rightinvariant) action of L.Besides, the braiding R coming in the definition of the algebra B(L, M) is taken to be a Hecke symmetry. This means that it is subject to the equationwhere q ∈ K is assumed to be generic. In particular, such a Hecke symmetry comes from the QG U q (sl(m)). This Hecke symmetry and all related objects will be called standard. The RTT and RE algebras associated with a standard Hecke symmetry are deformations of the algebra Sym(gl(m)). With the use of other Hecke symmetries we can get analogous deformations of the super-algebra Sym(gl(m|n)).Note that any RE algebra associated with a Hecke symmetry has another basis in which the permutation relations between basic elements are quadratic-linear. So, the RE algebra in this basis becomes more similar to the enveloping algebra U (gl(m|n)). We call this quadraticlinear algebra the modified RE (mRE) algebra 3 . Namely in this form the RE algebra comes in constructing quantum analogs of vector fields.Besides, in [GPS2] we have con...
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