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Pre-Calabi-Yau algebras as noncommutative Poisson structures

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<mark>Journal publication date</mark>1/02/2021
<mark>Journal</mark>Journal of Algebra
Volume567
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 [14] 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
. 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 emphasises the special role of the fourth component of 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
for any associative algebra A.