  Let R: V x V -> V x V be a Hecke type solution of the quantum Yang-Baxter
equation (a Hecke symmetry). Then, the Hilbert-Poincre' series of the
associated R-exterior algebra of the space V is a ratio of two polynomials of
degree m (numerator) and n (denominator). Assuming R to be skew-invertible, we
define a rigid quasitensor category SW(V) of vector spaces, generated by the
space V and its dual V*, and compute certain numerical characteristics of its
objects. Besides, we introduce a braided bialgebra structure in the modified
Reflection Equation Algebra, associated with R, and equip objects of the
category SW(V) with an action of this algebra. In the case related to the
quantum group U_q(sl(m)), we consider the Poisson counterpart of the modified
Reflection Equation Algebra and compute the semiclassical term of the pairing,
defined via the categorical (or quantum) trace.
