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Algebraic structures connected with pairs of compatible associative algebras

Alexander Odesskii·Vladimir Sokolov
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Abstract

We study associative multiplications in semi-simple associative algebras over C compatible with the usual one or, in other words, linear deformations of semi-simple associative algebras over C. It turns out that these deformations are in one-to-one correspondence with representations of certain algebraic structures, which we call M-structures in the matrix case and PM-structures in the case of direct sums of several matrix algebras. We also investigate various properties of PM-structures, provide numerous examples and describe an important class of PM-structures. The classification of these PM-structures naturally leads to affine Dynkin diagrams of A, D, E-type.

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Alexander Odesskii, Vladimir Sokolov. 2006-02-02. Algebraic structures connected with pairs of compatible associative algebras. https://arxiv.org/abs/math/0512499

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