Weyl approach to representation theory of reflection equation algebra

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 R-matrix 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)=Tr_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.Comment: LaTeX2e file, 27 pages, no figure

q-Index on braided non-commutative spheres

To some Hecke symmetries (i.e. Yang-Baxter braidings of Hecke type) we assign algebras called braided non-commutative spheres. For any such algebra, we introduce and compute a q-analog of the Chern-Connes index. Unlike the standard Chern-Connes index, ours is based on the so-called categorical trace specific for a braided category in which the algebra in question is represented.Comment: Essentially revised version. Latex file, 22 pages, no figure