Dual Hodge decompositions and derived Poisson brackets

TL;DR

This paper explores Hodge decompositions in cyclic and Hochschild homology of universal enveloping algebras of Lie algebras, linking algebraic structures with topological interpretations and Poisson brackets.

Contribution

It generalizes operadic cyclic homology constructions and connects Hodge decompositions with topological and Poisson structures, providing new insights into string topology.

Findings

Hodge decompositions relate to $S^1$-equivariant homology of free loop spaces.

The derived Poisson structure preserves the Hodge filtration.

Chas-Sullivan Lie algebra admits a natural Hodge filtration.

Abstract

We study general properties of Hodge-type decompositions of cyclic and Hochschild homology of universal enveloping algebras of (DG) Lie algebras. Our construction generalizes the operadic construction of cyclic homology of Lie algebras due to Getzler and Kapranov. We give a topological interpretation of such Lie Hodge decompositions in terms of $S^1$-equivariant homology of the free loop space of a simply connected topological space. We prove that the canonical derived Poisson structure on a universal enveloping algebra arising from a cyclic pairing on the Koszul dual coalgebra preserves the Hodge filtration on cyclic homology. As an application, we show that the Chas-Sullivan Lie algebra of any simply connected closed manifold carries a natural Hodge filtration. We conjecture that the Chas-Sullivan Lie algebra is actually graded, i.e. the string topology bracket preserves the Hodge decomposition.