Representation theory of (modified) Reflection Equation Algebra of GL(m|n) type

Abstract

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.