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Journal Article

#### Pre-Calabi-Yau algebras as noncommutative Poisson structures

##### MPS-Authors

##### External Ressource

https://doi.org/10.1016/j.jalgebra.2020.08.029

(Publisher version)

##### Fulltext (public)

arXiv:1906.07134.pdf

(Preprint), 222KB

##### Supplementary Material (public)

There is no public supplementary material available

##### 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 $A$.