Hodge decomposition of string topology

Let X be a simply connected closed oriented manifold of rationally elliptic homotopy type. We prove that the string topology bracket on the S-1-equivariant homology (H) over bar (S1)(*)(LX, Q) of the free loop space of X preserves the Hodge decomposition of (H) over bar (S1)(*)(LX, Q), making it a bigraded Lie algebra. We deduce this result from a general theorem on derived Poisson structures on the universal enveloping algebras of homologically nilpotent finite-dimensional DG Lie algebras. Our theorem settles a conjecture of [7].