Around Van den Bergh s double brackets for different bimodule structures.pdf (2.84 MB)
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posted on 2023-03-02, 13:43 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
Funding
University of Glasgow
Loughborough University
History
School
- Science
Department
- Mathematical Sciences
Published in
Communications in AlgebraVolume
51Issue
4Pages
1673-1706Publisher
Taylor & FrancisVersion
- VoR (Version of Record)
Rights holder
©The Author(s)Publisher statement
This is an Open Access article distributed under the terms of the Creative Commons Attribution-NonCommercial License (http://creativecommons.org/licenses/by-nc/4.0/), which permits unrestricted non-commercial use, distribution, and reproduction in any medium, provided the original work is properly cited.Acceptance date
2022-10-16Publication date
2022-11-09Copyright date
2022ISSN
0092-7872eISSN
1532-4125Publisher version
Language
- en