# Pre-Calabi-Yau algebras and double Poisson brackets

## TL;DR

This paper reveals how double Poisson algebras are embedded within pre-Calabi-Yau structures, establishing a link that induces Poisson brackets on representation spaces of associative algebras.

## Contribution

It explicitly connects double Poisson algebras to pre-Calabi-Yau structures, highlighting the role of the fourth component in this relationship.

## Key findings

- Double Poisson algebra structures correspond to specific parts of pre-Calabi-Yau solutions.
- Pre-Calabi-Yau structures induce Poisson brackets on representation spaces.
- The results apply to any associative algebra, emphasizing the generality of the approach.

## 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 $A$.