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Pre-Calabi-Yau algebras as noncommutative Poisson structures

MPS-Authors
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Iyudu,  Natalia
Max Planck Institute for Mathematics, Max Planck Society;

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Kontsevich,  Maxim
Max Planck Institute for Mathematics, Max Planck Society;

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Vlassopoulos,  Yannis
Max Planck Institute for Mathematics, Max Planck Society;

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Fulltext (public)

arXiv:1906.07134.pdf
(Preprint), 222KB

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Citation

Iyudu, N., Kontsevich, M., & Vlassopoulos, Y. (2021). Pre-Calabi-Yau algebras as noncommutative Poisson structures. Journal of Algebra, 567, 63-90. doi:10.1016/j.jalgebra.2020.08.029.


Cite as: http://hdl.handle.net/21.11116/0000-0007-A23C-F
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 $A$.