Pre-Calabi-Yau algebras as noncommutative Poisson structures

We give an explicit formula showing how the double Poisson algebra introduced in \cite{VdB} appears as a particular part of pre-Calabi-Yau structure, i.e. cyclically invariant, with respect to natural inner form, solution Maurer-Cartan equation on $A\oplus A^*$. Specific this is described, which one-to-one correspondence structures. The result holds for any associative $A$ and emphasizes special role fourth component structure respect. As consequence we have that appropriate structures induce brackets representation spaces $({\rm Rep}_n A)^{Gl_n}$ $A$.