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.

It is known that every quasitriangular Hopf algebra H can be converted by a process of transmutation into a braided group B(H, H). The latter is a certain braided-cocommutative Hopf algebra in the braided monoidal category of //-modules. We use this transmutation construction to relate two approaches to the quantization of enveloping algebras.Specifically, we compute B(H, H) in the case when H is the quasitriangular Hopf algebra (quantum group) obtained by Drinfeld's twisting construction on a cocommutative Hopf algebra H. In the case when H is triangular we recover the S-Hopf algebra H F previously obtained as a deformation-quantization of H. Here Hf is a Hopf algebra in a symmetric monoidal category. We thereby extend the definition of H F to the braided case where H is strictly quasitriangular. We also compute its structure to lowest order in a quantization parameter h. In this way we show that B(U q (g), U q (g)) is the quantization of a certain generalized Poisson bracket associated to the Drinfeld-Jimbo solution of the classical Yang-Baxter equations.

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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.