Hodge decomposition of string topology
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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].
2019 ◽
Vol 15
(1)
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pp. 167-183
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2007 ◽
Vol 06
(01)
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pp. 49-69
2007 ◽
Vol 5
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pp. 195-200
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2016 ◽
Vol 2016
(716)
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Keyword(s):
2007 ◽
Vol 17
(03)
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pp. 527-555
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