Double Poisson brackets on free associative algebras

TL;DR

This paper explores double Poisson structures on free associative algebras, focusing on quadratic brackets, their relations to associative Yang-Baxter equations, and classifies quadratic brackets for algebras with two generators, providing new examples.

Contribution

It introduces a classification of quadratic double Poisson brackets on free algebras with two generators and presents new examples, linking them to associative Yang-Baxter equations.

Findings

Established relations between double Poisson brackets and associative Yang-Baxter equations.

Classified quadratic double Poisson brackets for two-generator free algebras.

Proposed numerous new examples of quadratic double Poisson brackets.

Abstract

We discuss double Poisson structures in sense of M. Van den Bergh on free associative algebras focusing on the case of quadratic Poisson brackets. We establish their relations with an associative version of Young-Baxter equations, we study a bi-hamiltonian property of the linear-quadratic pencil of the double Poisson structure and propose a classification of the quadratic double Poisson brackets in the case of the algebra with two free generators. Many new examples of quadratic double Poisson brackets are proposed.