Representation theory of (modified) reflection equation algebra of $GL(m\vert n)$ type

Let $R:V^{\otimes 2}\to V^{\otimes 2}$ be a Hecke type solution of the quantum Yang-Baxter 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 ${\rm SW}(V_{(m\vert n)})$ 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 ${\rm SW}(V_{(m\vert n)})$ 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.