Double Poisson brackets on low dimensional algebras

TL;DR

This paper classifies double Poisson brackets on finite-dimensional algebras, showing they are mostly inner, and explores their structures on matrix and triangular matrix algebras, with implications for representation spaces.

Contribution

It proves that all double Poisson brackets on matrix and semisimple algebras are inner, and provides explicit descriptions for 2x2 upper triangular matrices.

Findings

All double Poisson brackets on matrix algebras are inner.

Double Poisson brackets on semisimple algebras are inner.

Explicit description of brackets on 2x2 upper triangular matrices.

Abstract

In this paper, we describe double Poisson brackets in the sense of M. Van den Bergh on certain finite-dimensional algebras. In particular we prove that all possible double Poisson brackets on matrix algebras are "inner", i.e. given by some commutators in bimodules. As a corollary of this result, we see that all possible double Poisson brackets in any finite-dimensional semisimple algebras over algebraically closed fields are also given by inner derivations. We further give a description of all double Poisson brackets on the algebra of 2x2 upper triangular matrices. We further discuss Poisson structures induced from the double Poisson brackets in its representation spaces of rank two and three. In the appendix, we describe modified double Poisson brackets (in the sense of S. Arthamonov) on this algebra.