# Pre-Calabi-Yau algebras and double Poisson brackets

@article{Iyudu2019PreCalabiYauAA, title={Pre-Calabi-Yau algebras and double Poisson brackets}, author={Natalia Iyudu and Maxim Kontsevich and Yannis Vlassopoulos}, journal={arXiv: Rings and Algebras}, year={2019} }

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…

## 3 Citations

### Pre-Calabi-Yau algebras and noncommutative calculus on higher cyclic Hochschild cohomology

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We prove $L_{\infty}$-formality for the higher cyclic Hochschild complex $\chH$ over free associative algebra or path algebra of a quiver. The $\chH$ complex is introduced as an appropriate tool for…

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We explain our previous results about Hochschild actions [Ka07,Ka08a] pertaining in particular to the co-product which appeared in a different form in [GH09] and provide a fresh look at the results.…

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In this article, we prove that double quasi-Poisson algebras, which are noncommutative analogues of quasi-Poisson manifolds, naturally give rise to pre-Calabi-Yau algebras. This extends one of the…

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