Integrable systems of interacting tops and $R$-matrix identities

We make a review of a wide class of integrable systems — the systems of interacting tops, which includes many-body systems and spin chains as particular examples. An important and useful tool for description of these models is given by $R$-matrices, which satisfy not only the quantum Yang-Baxter equation but also a quadratic relation known as the associative Yang-Baxter equation. Due to this property $R$-matrices satisfy also a number of relations (identities), which could be understood as matrix generalizations of elliptic functions identities. Using this approach, we will describe a construction of quantum interacting tops. The Hamiltonians for these models are given by matrix (spin) generalizations of Ruijsenaars-Macdonald operators. Constructions of new families of integrable systems will be discussed as well. In particular, the long-range anisotropic quantum spin chains of Haldane-Shastry type will be obtained.