Parameter-dependent associative Yang-Baxter equations and Poisson brackets

TL;DR

This paper explores parameter-dependent associative Yang-Baxter equations and their role in describing a broad class of parameter-dependent Poisson structures, providing a classification for one-dimensional solutions.

Contribution

It introduces associative analogues of classical Yang-Baxter equations with parameters and classifies all solutions in one dimension.

Findings

Associative Yang-Baxter equations relate to Poisson structures.

Classification of one-dimensional solutions provided.

New connections between associative equations and Poisson geometry.

Abstract

We discuss associative analogues of classical Yang-Baxter equation meromorphically dependent on parameters. We discover that such equations enter in a description of a general class of parameter-dependent Poisson structures and double Lie and Poisson structures in sense of M. Van den Bergh. We propose a classification of all solutions for one-dimensional associative Yang-Baxter equations.