Higher Brackets on Cyclic and Negative Cyclic (Co)Homology

Abstract The purpose of this article is to embed the string topology bracket developed by Chas–Sullivan and Menichi on negative cyclic cohomology groups as well as the dual bracket found by de Thanhoffer de Völcsey–Van den Bergh on negative cyclic homology groups into the global picture of a noncommutative differential (or Cartan) calculus up to homotopy on the (co)cyclic bicomplex in general, in case a certain Poincaré duality is given. For negative cyclic cohomology, this in particular leads to a Batalin–Vilkoviskiĭ (BV) algebra structure on the underlying Hochschild cohomology. In the special case in which this BV bracket vanishes, one obtains an $e_3$-algebra structure on Hochschild cohomology. The results are given in the general and unifying setting of (opposite) cyclic modules over (cyclic) operads.

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The BV-Algebra Structure on the Hochschild Cohomology of Local Algebras of Quaternion Type in Characteristic 2

Journal of Mathematical Sciences ◽

10.1007/s10958-016-3118-1 ◽

2016 ◽

Vol 219 (3) ◽

pp. 427-461

Author(s):

A. A. Ivanov

Keyword(s):

Hochschild Cohomology ◽

Algebra Structure ◽

Local Algebras ◽

Quaternion Type ◽

Characteristic 2 ◽

Bv Algebra

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THE BV-ALGEBRA STRUCTURE OF THE HOCHSCHILD COHOMOLOGY OF THE GROUP RING OF CYCLIC GROUPS OF PRIME ORDER

Geometric, Algebraic and Topological Methods for Quantum Field Theory ◽

10.1142/9789814730884_0009 ◽

2016 ◽

Author(s):

Andrés Angel ◽

Diego Duarte

Keyword(s):

Group Ring ◽

Prime Order ◽

Hochschild Cohomology ◽

Algebra Structure ◽

Cyclic Groups ◽

Bv Algebra

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A Cohomological Property of π-invariant Elements

Canadian Mathematical Bulletin ◽

10.4153/cmb-2011-184-7 ◽

2013 ◽

Vol 56 (3) ◽

pp. 534-543

Author(s):

M. Filali ◽

M. Sangani Monfared

Keyword(s):

Hilbert Space ◽

Banach Algebra ◽

Orthonormal Basis ◽

Hochschild Cohomology ◽

Separable Hilbert Space ◽

Continuous Representation ◽

Cohomology Groups ◽

Cohomological Property ◽

Coordinate Functions ◽

Special Case

Abstract.Let A be a Banach algebra and let be a continuous representation of A on a separable Hilbert space H with dim H = m. Let πi j be the coordinate functions of π with respect to an orthonormal basis and suppose that for each and . Under these conditions, we call an element left π-invariant if In this paper we prove a link between the existence of left π-invariant elements and the vanishing of certain Hochschild cohomology groups of A. Our results extend an earlier result by Lau on F-algebras and recent results of Kaniuth, Lau, Pym, and and the second author in the special case where π : A → C is a non-zero character on A.

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Excision in Banach simplicial and cyclic cohomology

Proceedings of the Edinburgh Mathematical Society ◽

10.1017/s0013091500019751 ◽

1998 ◽

Vol 41 (2) ◽

pp. 411-427 ◽

Cited By ~ 2

Author(s):

Zinaida A. Lykova

Keyword(s):

Exact Sequence ◽

Banach Algebras ◽

Approximate Identity ◽

Cyclic Cohomology ◽

Cohomology Groups ◽

Continuous Version ◽

Homology Groups ◽

Exact Sequences

We prove that, for every extension of Banach algebras 0 → B →A → D → 0 such that B has a left or right bounded approximate identity, the existence of an associated long exact sequence of Banach simplicial or cyclic cohomology groups is equivalent to the existence of one for homology groups. It follows from the continuous version of a result of Wodzicki that associated long exact sequences exist. In particular, they exist for every extension of C*-algebras.

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Hochschild cohomology of tensor products of topological algebras

Proceedings of the Edinburgh Mathematical Society ◽

10.1017/s0013091508001065 ◽

2010 ◽

Vol 53 (2) ◽

pp. 447-470

Author(s):

Zinaida A. Lykova

Keyword(s):

Hochschild Cohomology ◽

Tensor Products ◽

Cyclic Cohomology ◽

Fréchet Spaces ◽

Cohomology Groups ◽

Topological Algebras ◽

Kunneth Formula

AbstractWe describe explicitly the continuous Hochschild and cyclic cohomology groups of certain tensor products of $\widehat{\otimes}$-algebras which are Fréchet spaces or nuclear DF-spaces. To this end we establish the existence of topological isomorphisms in the Künneth formula for the cohomology of complete nuclear DF-complexes and in the Künneth formula for continuous Hochschild cohomology of nuclear $\widehat{\otimes}$-algebras which are Fréchet spaces or DF-spaces for which all boundary maps of the standard homology complexes have closed ranges.

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A new description of equivariant cohomology for totally disconnected groups

Journal of K-theory K-theory and its Applications to Algebra Geometry and Topology ◽

10.1017/is007011019jkt020 ◽

2008 ◽

Vol 1 (3) ◽

pp. 431-472 ◽

Cited By ~ 1

Author(s):

Christian Voigt

Keyword(s):

Equivariant Cohomology ◽

Simplicial Complexes ◽

Cyclic Homology ◽

Cohomology Theory ◽

Cohomology Groups ◽

Natural Class ◽

New Approach ◽

Totally Disconnected

AbstractWe consider smooth actions of totally disconnected groups on simplicial complexes and compare different equivariant cohomology groups associated to such actions. Our main result is that the bivariant equivariant cohomology theory introduced by Baum and Schneider can be described using equivariant periodic cyclic homology. This provides a new approach to the construction of Baum and Schneider as well as a computation of equivariant periodic cyclic homology for a natural class of examples. In addition we discuss the relation between cosheaf homology and equivariant Bredon homology. Since the theory of Baum and Schneider generalizes cosheaf homology we finally see that all these approaches to equivariant cohomology for totally disconnected groups are closely related.

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Twisted Homology of Quantum SL(2) - Part II

Journal of K-theory K-theory and its Applications to Algebra Geometry and Topology ◽

10.1017/is009009022jkt091 ◽

2009 ◽

Vol 6 (1) ◽

pp. 69-98 ◽

Cited By ~ 5

Author(s):

Tom Hadfield ◽

Ulrich Krähmer

Keyword(s):

Noncommutative Geometry ◽

Hochschild Cohomology ◽

Cyclic Homology ◽

Coordinate Ring ◽

Volume Form ◽

Twisted Homology

AbstractWe complete the calculation of the twisted cyclic homology of the quantised coordinate ring = ℂq [SL(2)] of SL(2) that we began in [14]. In particular, a nontrivial cyclic 3-cocycle is constructed which also has a nontrivial class in Hochschild cohomology and thus should be viewed as a noncommutative geometry analogue of a volume form.

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Comparison morphisms between two projective resolutions of monomial algebras

Revista de la Unión Matemática Argentina ◽

10.33044/revuma/v59n1a01 ◽

2017 ◽

pp. 1-31

Author(s):

María Julia Redondo ◽

Lucrecia Román

Keyword(s):

Lie Algebra ◽

Hochschild Cohomology ◽

Cohomology Groups ◽

Efficient Tool ◽

Simple Description ◽

Monomial Algebra ◽

Cup Product ◽

Lie Bracket ◽

Projective Resolutions ◽

Bar Resolution

We construct comparison morphisms between two well-known projective resolutions of a monomial algebra $A$: the bar resolution $\operatorname{\mathbb{Bar}} A$ and Bardzell's resolution $\operatorname{\mathbb{Ap}} A$; the first one is used to define the cup product and the Lie bracket on the Hochschild cohomology $\operatorname{HH} ^*(A)$ and the second one has been shown to be an efficient tool for computation of these cohomology groups. The constructed comparison morphisms allow us to show that the cup product restricted to even degrees of the Hochschild cohomology has a very simple description. Moreover, for $A= \mathbb{k} Q/I$ a monomial algebra such that $\dim_ \mathbb{k} e_i A e_j = 1$ whenever there exists an arrow $\alpha: i \to j \in Q_1$, we describe the Lie action of the Lie algebra $\operatorname{HH}^1(A)$ on $\operatorname{HH}^{\ast} (A)$.

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