Pre-Calabi-Yau algebras and double Poisson brackets

Iyudu, Natalia and Kontsevich, Maxim and Vlassopoulos, Yannis (2021) Pre-Calabi-Yau algebras and double Poisson brackets. Journal of Algebra. pp. 63-90. ISSN 0021-8693

[thumbnail of 1906.07134v1]
Text (1906.07134v1)
1906.07134v1.pdf - Accepted Version
Available under License Creative Commons Attribution Non-commercial No Derivatives.

Download (226kB)


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 $A$.

Item Type:
Journal Article
Journal or Publication Title:
Journal of Algebra
Additional Information:
This is the author’s version of a work that was accepted for publication in Journal of Algebra. Changes resulting from the publishing process, such as peer review, editing, corrections, structural formatting, and other quality control mechanisms may not be reflected in this document. Changes may have been made to this work since it was submitted for publication. A definitive version was subsequently published inJournal of Algebra, ?, ?, 2021 DOI: 10.1016/j.jalgebra.2020.08.029
Uncontrolled Keywords:
?? math.ra16a22, 16s37, 16y99algebra and number theory ??
ID Code:
Deposited By:
Deposited On:
08 Jun 2021 15:15
Last Modified:
02 Jun 2024 23:53