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

Fairon, M. and McCulloch, C. (2023) Around Van den Bergh’s double brackets for different bimodule structures. Communications in Algebra, 51(4), pp. 1673-1706. (doi: 10.1080/00927872.2022.2140349)

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 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.

• Item Type: Articles
• Additional Information: The research of the first author was partly supported by a Rankin-Sneddon Research Fellowship of the University of Glasgow, and a Doctoral Prize Fellowship of Loughborough University. The second author was partly supported by a Summer Project Bursary of the University of Glasgow.
• Status: Published
• Refereed: Yes
• Glasgow Author(s) Enlighten ID: McCulloch, Mr Colin and Fairon, Dr Maxime
• Authors: Fairon, M., and McCulloch, C.
• College/School: College of Science and Engineering > School of Mathematics and Statistics; College of Science and Engineering > School of Mathematics and Statistics > Mathematics
• Journal Name: Communications in Algebra
• Publisher: Taylor & Francis
• ISSN: 0092-7872
• ISSN (Online): 1532-4125
• Published Online: 09 November 2022
• Copyright Holders: Copyright © 2022 The Authors
• First Published: First published in Communications in Algebra 51(4):1673-1706
• Publisher Policy: Reproduced under a Creative Commons License