Double Poisson brackets on free associative algebras

Alexander Odesskii (1) , Vladimir Rubtsov (2) , Vladimir Sokolov (1)

1 Chercheur indépendant
2 LAREMA - Laboratoire Angevin de Recherche en Mathématiques

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.

Mots clés: double Poisson structures; associative Yang-Baxter equations

Domaines: Physique mathématique [math-ph]; Physique mathématique [math-ph]; Algèbres quantiques [math.QA]; Théorie des représentations [math.RT]

Alexander Odesskii, Vladimir Rubtsov, Vladimir Sokolov. Double Poisson brackets on free associative algebras. 2012. ⟨hal-00725993⟩