  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.
