Hodge decomposition of string topology

Abstract

Abstract 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 $ {\\overline {\\text {H}}}_\\ast ^{S^1}({\\mathcal {L}} X,{\\mathbb {Q}}) $ of the free loop space of X preserves the Hodge decomposition of $ {\\overline {\\text {H}}}_\\ast ^{S^1}({\\mathcal {L}} X,{\\mathbb {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].