PreCalabiYau 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 preCalabiYau structure, i.e. cyclically invariant, with respect to the natural inner form, solution of the MaurerCartan equation on $A\oplus A^*$. Specific part of this solution is described, which is in onetoone 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 preCalabiYau structure in this respect. As a consequence we have that appropriate preCalabiYau structures induce a Poisson brackets on representation spaces $({\rm Rep}_n A)^{Gl_n}$ for any associative algebra $A$.
 Publication:

arXiv eprints
 Pub Date:
 June 2019
 arXiv:
 arXiv:1906.07134
 Bibcode:
 2019arXiv190607134I
 Keywords:

 Mathematics  Rings and Algebras;
 16A22;
 16S37;
 16Y99
 EPrint:
 J. Algebra, 2020