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

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 $\mathrm{Rep}(A,n)$ 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.

34 pages, 1 figure. Comments are welcome

References in corpus (3)

• Morphisms of double (quasi-)Poisson algebras and action-angle duality of integrable systems
• Hamiltonian structures for integrable nonabelian difference equations
• Double Multiplicative Poisson Vertex Algebras