Anti-Frobenius algebras and associative Yang–Baxter equation

A. I. Zobnin

Lomonosov Moscow State University, Department of Mechanics and Mathematics, Moscow

Abstract: Associative Yang–Baxter equation arises in different areas of algebra, e.g., when studying double quadratic Poisson brackets, non-abelian quadratic Poisson brackets, or associative algebras with cyclic 2-cocycle (anti-Frobenius algebras). Precisely, faithful representations of anti-Frobenius algebras (up to isomorphism) are in one-to-one correspondence with skew-symmetric solutions of associative Yang–Baxter equation (up to equivalence). Following the work of Odesskii, Rubtsov and Sokolov and using computer algebra system Sage, we found some constant skew-symmetric solutions of associative Yang–Baxter equation and construct corresponded non-abelian quadratic Poisson brackets.

Keywords: associative Yang–Baxter equation, anti-Frobenius algebras, non-abelian quadratic Poisson brackets.

UDC: 512.552

Received: 21.03.2014

Language: English

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