This work deals with an algebro-geometric theory of solutions of the classical Yang-Baxter equation based on torsion free coherent sheaves of Lie algebras on Weierstraß cubic curves.

In this paper we study the combinatorics of quasi-trigonometric solutions of the classical Yang-Baxter equation, arising from simple vector bundles on a nodal Weierstraß cubic.

We describe a method for doing computations with Orlov's equivalence between the bounded derived category of certain hypersurfaces and the stable category of graded matrix factorisations of the polynomials describing these hypersurfaces. In the case of a smooth elliptic curve over an algebraically closed field we describe the indecomposable graded matrix factorisations of rank one. Since every indecomposable Maximal Cohen-Macaulay module over the completion of a smooth cubic curve is gradable, we obtain explicit descriptions of all indecomposable rank one matrix factorisations of smooth cubic potentials. Finally, we explain how to compute all indecomposable matrix factorisations of higher rank with the help of a computer algebra system.

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