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].

中文翻译：

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字符串拓扑的霍奇分解

让X是有理椭圆同伦型的单连通闭向流形。我们证明了字符串拓扑括号 $S^1$ -等变同源性 $ {\overline {\text {H}}}_\ast ^{S^1}({\mathcal {L}} X,{\mathbb {Q}}) $ 的自由循环空间X保留 Hodge 分解 $ {\overline {\text {H}}}_\ast ^{S^1}({\mathcal {L}} X,{\mathbb {Q}}) $ ，使其成为二分李代数。我们从关于同调幂零有限维 DG 李代数的通用包络代数上的派生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]的一个猜想。