Pre-Calabi-Yau algebras and double Poisson brackets

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 $A\oplus A^{\ast }$. 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 $({\mathrm{Rep}}_{n}A{)}^{G{l}_n}$ for any associative algebra $A$.

References in corpus (2)

• Pre-Calabi-Yau algebras and double Poisson brackets
• Pre-Calabi-Yau algebras and topological quantum field theories

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