Pre-Calabi--Yau algebras and homotopy double Poisson gebras

Leray, Johan; Vallette, Bruno

doi:10.48550/arxiv.2203.05062

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2022

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“…In particular, this principle emphasises an important reason for the development of the study of double Poisson brackets. Some other useful aspects of this theory include classification results [5,21,22,24,27], double Poisson cohomology [1,23,27], connection to (pre-)Calabi-Yau algebras [6,13,15,19], study from the point of view of properads [18,19] and relation to integrable systems [7,10,11,12].…”

Section: Introductionmentioning

confidence: 99%

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

Fairon¹,

McCulloch²

2022

Preprint

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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 formalisation of the widespread tensor notation used to write Poisson brackets of matrices in mathematical physics.

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