# Pre-Calabi-Yau algebras and double Poisson brackets

Natalia Iyudu, Maxim Kontsevich, Yannis Vlassopoulos

Research output: Working paper

## 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 $A$.
Original language English ArXiv 1-24 24 Published - 17 Jun 2019

## Keywords

• math.RA
• 16A22, 16S37, 16Y99