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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 $A$.