Supersymmetric generalization of q-deformed long-range spin chains of Haldane-Shastry type and trigonometric GL(N|M) solution of associative Yang-Baxter equation

We propose commuting sets of matrix-valued difference operators in terms of trigonometric GL(N|M)-valued R-matrices thus providing quantum supersymmetric (and possibly anisotropic) spin Ruijsenaars-Macdonald operators. Two types of trigonometric supersymmetric R-matrices are used for this purpose. The first is the one related to the affine quantized algebra Uˆq(gl(N|M)). The second is a graded version of the standard Zn-invariant An−1 type R-matrix. We show that being properly normalized the latter graded R-matrix satisfies the associative Yang-Baxter equation. Next, we discuss construction of long-range spin chains using the Polychronakos freezing trick. As a result we obtain a new family of spin chains, which extends the gl(N|M)-invariant Haldane-Shastry spin chain to q-deformed case with possible presence of anisotropy.