In this article, we develop a geometric method to construct solutions of the classical Yang-Baxter equation, attaching to the Weierstrass family of plane cubic curves and a pair of coprime positive integers, a family of classical r-matrices. It turns out that all elliptic r-matrices arise in this way from smooth cubic curves. For the cuspidal cubic curve, we prove that the obtained solutions are rational and compute them explicitly. We also describe them in terms of Stolin's classification and prove that they are degenerations of the corresponding elliptic solutions.

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In this paper we study unitary solutions of the associative Yang-Baxter equation (AYBE) with spectral parameters. We show that to each point τ from the upper half-plane and an invertible (n × n) matrix B with complex coefficients one can attach a solution of AYBE with values in Mat n×n (C) ⊗ Mat n×n (C), depending holomorphically on τ and B. Moreover, we compute some of these solutions explicitly.

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We give a complete description of the cluster-mutation classes of diagrams of Dynkin types A, B, D and of affine Dynkin types B (1) , C (1) , D (1) via certain families of diagrams.