Set-theoretical solutions to the Yang-Baxter Relation from factorization of matrix polynomials and θ\theta-functions

New set-theoretical solutions to the Yang-Baxter Relation are constructed. These solutions arise from the decompositions "in different order" of matrix polynomials and $\theta$-functions. We also construct a "local action of the symmetric group" in these cases, generalizations of the action of the symmetric group $S_N$ given by the set-theoretical solution.Comment: 9 pages, to appear in Moscow Math Journa

Compatible quadratic Poisson brackets related to a family of elliptic curves

We construct nine pairwise compatible quadratic Poisson structures such that a generic linear combination of them is associated with an elliptic algebra in n generators. Explicit formulas for Casimir elements of this elliptic Poisson structure are obtained.Comment: 17 pages, Latex, major change