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 $\mathrm{G}\mathrm{L}(N\mathrm{\mid }M)$-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 ${\widehat{\mathcal{U}}}_q(\mathrm{g}\mathrm{l}(N\mathrm{\mid }M))$. The second is a graded version of the standard ${\mathbb{Z}}_n$-invariant $A_{n-1}$ 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 $\mathrm{g}\mathrm{l}(N\mathrm{\mid }M)$-invariant Haldane-Shastry spin chain to q-deformed case with possible presence of anisotropy.