Higher rank generalization of 11-vertex rational R-matrix: IRF-Vertex relations and associative Yang-Baxter equation

We study ${\mathrm{G}\mathrm{L}}_N$ rational $R$-matrix, which turns into the 11-vertex $R$-matrix in the $N=2$ case. First, we describe its relations to dynamical and semi-dynamical $R$-matrices using the IRF-Vertex type transformations. As a by-product a new explicit form for ${\mathrm{G}\mathrm{L}}_N$ $R$-matrix is derived. Next, we prove the quantum and the associative Yang-Baxter equations. A set of other $R$-matrix properties and $R$-matrix identities are proved as well.