Cyclic pairings and derived Poisson structures

TL;DR

This paper explores a canonical derived Poisson structure on universal enveloping algebras of DG Lie algebras, linking it to string topology and representation homology through cyclic pairings.

Contribution

It introduces a new understanding of how derived characters interact with the Poisson structures on universal enveloping algebras and representation homology.

Findings

Established a canonical derived Poisson structure on $ ext{U} ext{a}$.

Demonstrated the intertwining of derived characters with Poisson structures.

Extended results to associative algebras.

Abstract

There is a canonical derived Poisson structure on the universal enveloping algebra $\mathcal{U}\mathfrak{a}$ of a (DG) Lie algebra $\mathfrak{a}$ that is Koszul dual to a cyclic cocommutative (DG) coalgebra. Interesting special cases of this derived Poisson structure include (an analog of) the Chas-Sullivan bracket on string topology. We study how certain derived character of $\mathfrak{a}$ intertwine this derived Poisson structure with the induced Poisson structure on the representation homology of $\mathfrak{a}$. In addition, we obtain an analog of one of our main results for associative algebras.