Around Van den Bergh's double brackets for different bimodule structures

TL;DR

This paper explores how different bimodule structures influence Van den Bergh's double brackets, affecting the induced Poisson structures on representation spaces and formalizing tensor notation in mathematical physics.

Contribution

It analyzes the impact of changing bimodule structures on double brackets and describes how these choices determine Poisson brackets and Jacobi identities.

Findings

Different bimodule structures fix analogues of Jacobi identity.

Induced Poisson brackets depend on bimodule choices.

Formalization of tensor notation for Poisson brackets in physics.

Abstract

A double Poisson bracket, in the sense of M. Van den Bergh, is an operation on an associative algebra $A$ which induces a Poisson bracket on each representation space $\operatorname{Rep}(A,n)$ in an explicit way. In this note, we study the impact of changing the Leibniz rules underlying a double bracket. This change amounts to make a suitable choice of $A$-bimodule structure on $A\otimes A$. In the most important cases, we describe how the choice of $A$-bimodule structure fixes an analogue to Jacobi identity, and we obtain induced Poisson brackets on representation spaces. The present theory also encodes a formalisation of the widespread tensor notation used to write Poisson brackets of matrices in mathematical physics.