Anti-Frobenius Algebras and Associative Yang-Baxter Equation

TL;DR

This paper explores the connection between anti-Frobenius algebras and solutions to the associative Yang-Baxter equation, using computational methods to find new solutions and related Poisson brackets.

Contribution

It identifies a correspondence between anti-Frobenius algebras and solutions to the associative Yang-Baxter equation, and constructs new non-abelian quadratic Poisson brackets using computer algebra.

Findings

Found new constant skew-symmetric solutions of the associative Yang-Baxter equation.

Constructed non-abelian quadratic Poisson brackets from these solutions.

Established a correspondence between anti-Frobenius algebras and solutions of the equation.

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.