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Dual Hodge decompositions and derived Poisson brackets

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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.

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Authors and Affiliations

  1. Department of Mathematics, Cornell University, Ithaca, NY, 14853-4201, USA

    Yuri Berest

  2. Department of Mathematics, Indiana University, Bloomington, IN, 47405, USA

    Ajay C. Ramadoss & Yining Zhang

Authors
  1. Yuri Berest
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  2. Ajay C. Ramadoss
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  3. Yining Zhang
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Correspondence to Yuri Berest.

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Berest, Y., Ramadoss, A.C. & Zhang, Y. Dual Hodge decompositions and derived Poisson brackets. Sel. Math. New Ser. 23, 2029–2070 (2017). https://doi.org/10.1007/s00029-017-0309-7

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