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Selecta Mathematica

, Volume 23, Issue 3, pp 2029–2070 | Cite as

Dual Hodge decompositions and derived Poisson brackets

  • Yuri BerestEmail author
  • Ajay C. Ramadoss
  • Yining Zhang
Article
  • 116 Downloads

Abstract

We study general properties of Hodge-type decompositions of cyclic and Hochschild homology of universal enveloping algebras of (DG) Lie algebras. Our construction generalizes the operadic construction of cyclic homology of Lie algebras due to Getzler and Kapranov. We give a topological interpretation of such Lie Hodge decompositions in terms of \(S^1\)-equivariant homology of the free loop space of a simply connected topological space. We prove that the canonical derived Poisson structure on a universal enveloping algebra arising from a cyclic pairing on the Koszul dual coalgebra preserves the Hodge filtration on cyclic homology. As an application, we show that the Chas–Sullivan Lie algebra of any simply connected closed manifold carries a natural Hodge filtration. We conjecture that the Chas–Sullivan Lie algebra is actually graded, i.e. the string topology bracket preserves the Hodge decomposition.

Mathematics Subject Classification

19D55 55P50 16E40 55P62 18G55 

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© Springer International Publishing 2017

Authors and Affiliations

  1. 1.Department of MathematicsCornell UniversityIthacaUSA
  2. 2.Department of MathematicsIndiana UniversityBloomingtonUSA

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