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Pairs of compatible associative algebras, classical Yang-Baxter equation and quiver representations

Alexander Odesskii·Vladimir Sokolov
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Abstract

Given an associative multiplication in matrix algebra compatible with the usual one or, in other words, linear deformation of matrix algebra, we construct a solution to the classical Yang-Baxter equation. We also develop a theory of such deformations and construct numerous examples. It turns out that these deformations are in one-to-one correspondence with representations of certain algebraic structures, which we call M-structures. We also describe an important class of M-structures related to the affine Dynkin diagrams of A, D, E-type. These M-structures and their representations are described in terms of quiver representations.

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Alexander Odesskii, Vladimir Sokolov. 2006-11-07. Pairs of compatible associative algebras, classical Yang-Baxter equation and quiver representations. https://arxiv.org/abs/math/0611200

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