Around Van den Bergh s double brackets for different bimodule structures.pdf (2.85 MB)
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journal contribution
posted on 2023-01-20, 11:34 authored by Maxime Fairon, Colin McCullochA 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 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⊗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 formalization of the widespread tensor notation used to write Poisson brackets of matrices in mathematical physics
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University of Glasgow
Loughborough University
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- Mathematical Sciences