Geodesic
Hodge decomposition of string topology
Forum of Mathematics, Sigma, Tome 9 (2021)

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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]. 
@article{10_1017_fms_2021_26,
     author = {Yuri Berest and Ajay C. Ramadoss and Yining Zhang},
     title = {Hodge decomposition of string topology},
     journal = {Forum of Mathematics, Sigma},
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     year = {2021},
     doi = {10.1017/fms.2021.26},
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Yuri Berest; Ajay C. Ramadoss; Yining Zhang. Hodge decomposition of string topology. Forum of Mathematics, Sigma, Tome 9 (2021). doi: 10.1017/fms.2021.26

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