  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.
