Batalin-Vilkovisky Algebras and Cyclic Cohomology of Hopf Algebras

Menichi, Luc

doi:10.1007/s10977-004-0480-4

Cited by 62 publications

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“…The second author proved in [51] that the Hochschild cohomology of a symmetric Frobenius algebra is a Batalin-Vilkovisky algebra. The group ring k[G] of a finite group G is one of the classical example of symmetric Frobenius algebras (Example 48 2)).…”

Section: Plan Of the Paper And Resultsmentioning

confidence: 99%

String topology of classifying spaces

Chataur

Menichi

2012

Journal Für Die Reine Und Angewandte Mathematik (Crelles Journal)

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Abstract. Let G be a finite group or a compact connected Lie group and let BG be its classifying space. Let LBG := map(S 1 , BG) be the free loop space of BG i.e. the space of continuous maps from the circle S 1 to BG. The purpose of this paper is to study the singular homology H * (LBG) of this loop space. We prove that when taken with coefficients in a field the homology of LBG is a homological conformal field theory. As a byproduct of our main theorem, we get a Batalin-Vilkovisky algebra structure on the cohomology H * (LBG). We also prove an algebraic version of this result by showing that the Hochschild cohomology HH

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Section: Plan Of the Paper And Resultsmentioning

confidence: 99%

String topology of classifying spaces

Chataur

Menichi

2012

Journal Für Die Reine Und Angewandte Mathematik (Crelles Journal)

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“…This theorem has been reproved and extended by many people [5,8,18,20,21,22,23,28] (in chronological order). The last proof, the proof of Eu et Schedler [8] looks similar to ours.…”

Section: Theorem 5 ([26 Example 215 and Theorem 31] (Corollary 19))mentioning

confidence: 98%

Batalin-Vilkovisky algebra structures on Hochschild cohomology

Menichi¹

2009

Bul. Soc. Math. France

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“…Full details of this result are given in Menichi [35]. This result is of course still true if A is a cyclic A 1 algebra; full details can be found in Tradler [41].…”

Section: The Cyclic Deligne Conjecturementioning

confidence: 87%

CyclicA_∞structures and Deligne’s conjecture

Ward

2012

Algebr. Geom. Topol.

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First we describe a class of homotopy Frobenius algebras via cyclic operads which we call cyclic A 1 algebras. We then define a suitable new combinatorial operad which acts on the Hochschild cochains of such an algebra in a manner which encodes the homotopy BV structure. Moreover we show that this operad is equivalent to the cellular chains of a certain topological (quasi)operad of CW-complexes whose constituent spaces form a homotopy associative version of the cacti operad of Voronov. These cellular chains thus constitute a chain model for the framed little disks operad, proving a cyclic A 1 version of Deligne's conjecture. This chain model contains the minimal operad of Kontsevich and Soibelman as a suboperad and restriction of the action to this suboperad recovers the results of Kontsevich and Soibelman [29] and Kaufmann and Schwell [27] in the unframed case. Additionally this proof recovers the work of Kaufmann in the case of a strict Frobenius algebra. We then extend our results to the context of cyclic A 1 categories, with an eye toward the homotopy BV structure present on the Hochschild cochains of the Fukaya category of a suitable symplectic manifold. 16E40, 18D50 IntroductionThroughout we let k be a field of characteristic ¤ 2. Given M , a closed oriented manifold, there are several meaningful constructions which associate a BV k -algebra to M including the (shifted) homology of the free loop space of M by Chas and Sullivan [6], the Hochschild cohomology of the singular cochains of M by Félix, Menichi and Thomas [13], and the symplectic cohomology of Cieliebak, Floer and Hofer [7] applied to the unit disk cotangent bundle of M (cf Seidel [36] which is expected to be cyclically invariant. We will call such a structure a cyclic A 1 category (see Definition 8.9). The cyclic invariance of the form would imply that the Hochschild cohomology of F.N / is a BV algebra and would endow the Hochschild cochains of F.N / with a homotopy BV structure. This is the homotopy BV structure that will be considered here in. As such, it would be expected that our principal objects of study would be cyclic A 1 categories and their Hochschild cohomology. However, as we will show, the study of cyclic A 1 categories and their Hochschild cohomology can be largely performed in the context of cyclic A 1 algebras and their Hochschild cohomology. As a result, for the sake of simplicity we conduct the bulk of our study in terms of cyclic A 1 algebras and conclude with the categorical generalization in Section 8.Cyclic A 1 algebras are a particular class of homotopy Frobenius algebras, namely those which relax the associativity to an A 1 algebra structure but do not resolve the bilinear form. Such algebras first appeared in Kontsevich [28]. Our approach will be to realize Frobenius algebras and more generally cyclic A 1 algebras as cyclic unital algebras over the cyclic unital operads As and A 1 . In particular we will take care to make the cyclic structure of the operad A 1 geometrically and combinatorially explicit. Summary of resul...

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Batalin-Vilkovisky Algebras and Cyclic Cohomology of Hopf Algebras

Cited by 62 publications

References 10 publications

String topology of classifying spaces

String topology of classifying spaces

Batalin-Vilkovisky algebra structures on Hochschild cohomology

CyclicA_∞structures and Deligne’s conjecture

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Batalin-Vilkovisky Algebras and Cyclic Cohomology of Hopf Algebras

Cited by 62 publications

References 10 publications

String topology of classifying spaces

String topology of classifying spaces

Batalin-Vilkovisky algebra structures on Hochschild cohomology

CyclicA∞structures and Deligne’s conjecture

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CyclicA_∞structures and Deligne’s conjecture