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  • 1906.07134v1

    Rights statement: This is the author’s version of a work that was accepted for publication in Journal of Algebra. Changes resulting from the publishing process, such as peer review, editing, corrections, structural formatting, and other quality control mechanisms may not be reflected in this document. Changes may have been made to this work since it was submitted for publication. A definitive version was subsequently published inJournal of Algebra, ?, ?, 2021 DOI: 10.1016/j.jalgebra.2020.08.029

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Pre-Calabi-Yau algebras and double Poisson brackets

Research output: Contribution to journalJournal articlepeer-review

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<mark>Journal publication date</mark>1/02/2021
<mark>Journal</mark>Journal of Algebra
Number of pages28
Pages (from-to)63-90
Publication StatusPublished
Early online date16/09/20
<mark>Original language</mark>English

Abstract

We give an explicit formula showing how the double Poisson algebra introduced in \cite{VdB} appears as a particular part of a pre-Calabi-Yau structure, i.e. cyclically invariant, with respect to the natural inner form, solution of the Maurer-Cartan equation on $A\oplus A^*$. Specific part of this solution is described, which is in one-to-one correspondence with the double Poisson algebra structures. The result holds for any associative algebra $A$ and emphasizes the special role of the fourth component of a pre-Calabi-Yau structure in this respect. As a consequence we have that appropriate pre-Calabi-Yau structures induce a Poisson brackets on representation spaces $({\rm Rep}_n A)^{Gl_n}$ for any associative algebra $A$.

Bibliographic note

This is the author’s version of a work that was accepted for publication in Journal of Algebra. Changes resulting from the publishing process, such as peer review, editing, corrections, structural formatting, and other quality control mechanisms may not be reflected in this document. Changes may have been made to this work since it was submitted for publication. A definitive version was subsequently published inJournal of Algebra, ?, ?, 2021 DOI: 10.1016/j.jalgebra.2020.08.029