Hodge decomposition of string topology

Berest, Yuri; Ramadoss, Ajay C.; Zhang, Yining

doi:10.1017/fms.2021.26

Cited by 5 publications

(2 citation statements)

References 51 publications

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“…Remark 3.3. The functor (3.23) was defined in [16] on a slightly larger -the so-called epicyclic -category ∆Ψ, which is an extension of ∆C describing the Adams operations on cyclic modules.…”

Section: Proof Straightforwardmentioning

confidence: 99%

Derived Character Maps of Groups Representations

Berest¹,

Ramadoss²

2022

Preprint

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In this paper, we construct and study derived character maps of finite-dimensional representations of ∞-groups. As models for ∞-groups we take homotopy simplicial groups, i.e. homotopy simplicial G op -algebras over the algebraic theory of groups (in the sense of [4]). We define cyclic, symmetric and representation homology for 'group algebras' k[Γ] over such groups and construct canonical trace maps relating these homology theories. In the case of one-dimensional representations, we show that our trace maps are of topological origin: they are induced by natural maps of (iterated) loop spaces that are well studied in homotopy theory. Using this topological interpretation, we deduce some algebraic results about representation homology: in particular, we prove that the symmetric homology of group algebras and one-dimensional representation homology are naturally isomorphic, provided the base ring k is a field of characteristic zero. We also study the behavior of the derived character maps of n-dimensional representations in the stable limit as n → ∞, in which case we show that they 'converge' to become isomorphisms.A comparison of these constructions can be found in [12, Appendix]. The relation to our present work is briefly discussed in the end of Section 3.2.

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Section: Proof Straightforwardmentioning

confidence: 99%

Derived Character Maps of Groups Representations

Berest¹,

Ramadoss²

2022

Preprint

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“…We refer the reader to [28, 4.5.4] for the operation. The result [3,Theorem 1.1] shows that the string bracket respects the Hodge decomposition in some sense. Thus, we are also interested in computations of string brackets, as described in 1.1 Problems, together with the consideration of the Hodge decomposition.…”

Section: The Cobar-type Emss and R-bv Exactnessmentioning

confidence: 99%

A reduction of the string bracket to the loop product

Kuribayashi,

Naito,

Wakatsuki

et al. 2021

Preprint

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The negative cyclic homology for a differential graded algebra over the rational field has a quotient of the Hochschild homology as a direct summand if the S-action is trivial. With this fact, we show that the string bracket in the sense of Chas and Sullivan is reduced to the loop product followed by the BV operator on the loop homology provided the given manifold is BV exact. The reduction is indeed derived from the equivalence between the BV exactness and the triviality of the S-action. Moreover, it is proved that a Lie bracket on the loop cohomology of the classifying space of a connected compact Lie group possesses the same reduction. By using these results, we consider the non-triviality of string brackets. Another highlight is that a simply-connected space with positive weights is BV exact. Furthermore, the higher BV exactness is also discussed featuring the cobar-type Eilenberg-Moore spectral sequence.

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Cartan calculus in string topology

Naito

2024

Proc. Amer. Math. Soc.

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In this paper, we investigate a Cartan calculus on the homology of free loop spaces which is introduced by Kuribayashi, Wakatsuki, Yamaguchi and the author [Cartan calculi on the free loop spaces, Preprint, arXiv:2207.05941, 2022]. In particular, it is proved that the Cartan calculus can be described by the loop product and the loop bracket in string topology. Moreover, by using the descriptions, we show that the loop product behaves well with respect to the Hodge decomposition of the homology of free loop spaces.

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Hodge decomposition of string topology

Cited by 5 publications

References 51 publications

Derived Character Maps of Groups Representations

Derived Character Maps of Groups Representations

A reduction of the string bracket to the loop product

Cartan calculus in string topology

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