Hodge decomposition of string topology

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 ${\overline{H}}_{\ast }^{S^1}(\mathcal{L}X,\mathbb{Q})$ of the free loop space of $X$ preserves the Hodge decomposition of ${\overline{H}}_{\ast }^{S^1}(\mathcal{L}X,\mathbb{Q})$ , 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 proposed in our earlier work.

28 pages

References in corpus (1)

• Free loop space and the cyclic bar construction

Cited by in corpus (2)

• Derived Character Maps of Groups Representations
• A reduction of the string bracket to the loop product