Pre-Calabi-Yau algebras and double Poisson brackets

Iyudu, Natalia and Kontsevich, Maxim and Vlassopoulos, Yannis (2021) Pre-Calabi-Yau algebras and double Poisson brackets. Journal of Algebra. pp. 63-90. ISSN 0021-8693

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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 $A$.