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 {\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].

摘要：设\(X\)是一个具有有理椭圆同伦型的单连通闭定向流形。我们证明了\(X\)的自由环路空间的\(S^1\)-等变同调\(\overline{\text{H}}_\ast^{S^1}(\mathcal{L}X,\mathbb{Q})\)上的弦拓扑括号保持\(\overline{\text{H}}_\ast^{S^1}(\mathcal{L}X,\mathbb{Q})\)的霍奇分解，使其成为一个双分次李代数。我们从一个关于同调幂零有限维\(DG\)李代数的泛包络代数上的导出泊松结构的一般定理推导出这个结果。我们的定理解决了[7]中的一个猜想。