1,721,491 research outputs found
Cyclic Gerstenhaber-Schack cohomology
Full text linkWe show that the diagonal complex computing the Gerstenhaber-Schack cohomology
of a bialgebra (that is, the cohomology theory governing bialgebra deformations)
can be given the structure of an operad with multiplication if the bialgebra is a (not
necessarily finite dimensional) Hopf algebra with invertible antipode; if the antipode is
involutive, the operad is even cyclic. Therefore, the Gerstenhaber-Schack cohomology of
any such Hopf algebra carries a Gerstenhaber resp. Batalin-Vilkoviski ̆ı algebra structure;
in particular, one obtains a cup product and a cyclic boundary B that generate the Gerstenhaber
bracket, and that allows to define cyclic Gerstenhaber-Schack cohomology. In case
the Hopf algebra in question is finite dimensional, the Gerstenhaber bracket turns out to
be zero in cohomology and hence the interesting structure is not given by this e2-algebra
structure but rather by the resulting e3-algebra structure, which is expressed in terms of
the cup product and B
Algèbres pré-Gerstenhaber à homotopie près
No full textThis paper is concerned by the concept of algebra up to homotopy for a structure defined by two operations and . An important example of such a structure is the Gerstenhaber algebra (commutatitve and Lie). The notion of Gerstenhaber algebra up to homotopy ( algebra) is known. Here, we give a definition of pre-Gerstenhaber algebra (pre-commutative and pre-Lie) allowing the construction of algebra. Given a structure of pre-commutative (Zinbiel) and pre-Lie algebra and working over the corresponding dual operads, we will give an explicit construction of the associated pre-Gerstenhaber algebra up to homotopy, this is a bicogebra (Leibniz and permutative) equipped with a codifferential which is a coderivation for the two coproducts
Algèbre Pré-Gerstenhaber à homotopie près
No full textInternational audienceRésumé. On étudie le concept d'algèbre à homotopie près pour une structure définie par deux opérations . et [ , ]. Un exemple important d'une telle structure est celui d'algèbre de Gerstenhaber (avec une structure commutative de degré 0 et une structure de Lie de degré −1). La notion d'algèbre de Gerstenhaber à homotopie près (G∞ algèbre) est connue : c'est une bicogèbre codifférentielle.Ici nous proposons une définition d'algèbre pré-Gerstenhaber (pré-commutative et pré-Lie) permettant la construction similaire d'une preG∞ algèbre.Partant d'une structure pré-commutative (Zinbiel) et pré-Lie, on utilise les opérades duales correspondantes, qui sont de Koszul. Nous donnons la construction explicite de l'algèbre à homotopie près associée. Celle-ci est une bicogèbre (Leibniz et permutative), munie d'une codifférentielle qui est une codérivation des deux coproduits.Abstract. This paper is concerned by the concept of algebra up to homotopy for a structure defined by two operations . and [ , ]. An important example of such a structure is the Gerstenhaber algebra (i.e. commutatitve structure with degree 0 and Lie structure with degree −1). The notion of Gerstenhaber algebra up to homotopy (G∞ algebra) is known: it is a codifferential bicogebra.Here, we give a definition of pre-Gerstenhaber algebra (pre-commutative and pre-Lie) allowing a similar construction for a preG∞ algebra.Given a structure of pre-commutative (Zinbiel) and pre-Lie algebra and working over the corresponding Koszul dual operads, we will give an explicit construction of the associated pre-Gerstenhaber algebra up to homotopy: it is a bicogebra (Leibniz and permutative) equipped with a codifferential which is a coderivation for the two coproducts
Algèbres et cogèbres de Gerstenhaber et cohomologie de Chevalley-Harrison
No full textThe fundamental example of Gerstenhaber algebra is the space of polyvector fields on , equipped with the wedge product and the Schouten bracket. In this paper, we explicitely describe what is the enveloping algebra of a Gerstenhaber algebra . This structure gives us a definition of the Chevalley-Harrison cohomology operator for . We finally show the nontriviality of a Chevalley-Harrison cohomology group for a natural Gerstenhaber subalgebra in
GERSTENHABER BRACKET ON HOPF ALGEBRA COHOMOLOGY
Full text linkM. A. Farinati, A. Solotar, and R. Taillefer showed that the Hopf algebra cohomology of a
quasi-triangular Hopf algebra, as a graded Lie algebra under the Gerstenhaber bracket, is abelian.
Motivated by the question of whether this holds for nonquasi-triangular Hopf algebras, we calculate
the Gerstenhaber bracket on Hopf algebra and Hochschild cohomologies of the Taft algebra
T_p for any integer p>2 which is a nonquasi-triangular Hopf algebra. We show that the bracket
is indeed zero on Hopf algebra cohomology of T_p, as in all known quasi-triangular Hopf algebras.
This example is the first known bracket computation for a nonquasi-triangular algebra.
We also show that Gerstenhaber brackets on Hopf algebra cohomology can be expressed via
an arbitrary projective resolution using Volkov���s homotopy liftings as generalized to some exact
monoidal categories. This is a special case of our more general result that a bracket operation on
cohomology is preserved under exact monoidal functors-one such functor is an embedding of
Hopf algebra cohomology into Hochschild cohomology. As a consequence, we show that this Lie
structure on Hopf algebra cohomology is abelian in positive degrees for all quantum elementary
abelian groups (T_p), most of which are nonquasi-triangular.
Also, we find a general formula for the bracket on Hopf algebra cohomology of any Hopf
algebra with bijective antipode on the bar resolution that is reminiscent of Gerstenhaber���s original
formula for Hochschild cohomology
Gerstenhaber brackets on Hochschild cohomology of general twisted tensor products
Full text linkWe present techniques for computing Gerstenhaber brackets on Hochschild cohomology of general twisted tensor product algebras. These techniques involve twisted tensor product resolutions and are based on recent results on Gerstenhaber brackets expressed on arbitrary bimodule resolutions.14 pages, small changes in the presentation, minor corrections, additional referee corrections, to appear in J. Pure Appl. Algebra. Corrections to three maps in Section
Monoidal categories and the Gerstenhaber bracket in Hochschild cohomology
No full textIn this monograph, the author extends S. Schwede's exact sequence interpretation of the Gerstenhaber bracket in Hochschild cohomology to certain exact and monoidal categories. Therefore the author establishes an explicit description of an isomorphism by A. Neeman and V. Retakh, which links \mathrm{Ext}-groups with fundamental groups of categories of extensions and relies on expressing the fundamental group of a (small) category by means of the associated Quillen groupoid. As a main result, the author shows that his construction behaves well with respect to structure preserving functors between exact monoidal categories. The author uses his main result to conclude, that the graded Lie bracket in Hochschild cohomology is an invariant under Morita equivalence. For quasi-triangular bialgebras, he further determines a significant part of the Lie bracket's kernel, and thereby proves a conjecture by L. Menichi. Along the way, the author introduces n-extension closed and entirely extension closed subcategories of abelian categories, and studies some of their properties
The Gerstenhaber-Schack complex for prestacks
No full textAbstract: The aim of this work is to construct a complex which through its higher structure directly controlls deformations of general prestacks, building on the work of Gerstenhaber and Schack for presheaves of algebras. In defining a Gerstenhaber-Schack complex C-GS(.) (A) for an arbitrary prestack A, we have to introduce a differential with an infinite sequence of components instead of just two as in the presheaf case. If (A)over-bar denotes the Grothendieck construction of A, which is a U-graded category, we explicitly construct inverse quasi-isomorphisms F and G between C-GS(.) (A) and the Hochschild complex C-u((A)over-bar), as well as a concrete homotopy T : FG -> 1, which had not been obtained even in the presheaf case. As a consequence, by applying the Homotopy Transfer Theorem, one can transfer the dg Lie structure present on the Hochschild complex in order to obtain an L-infinity-structure on C-GS(.) (A), which controlls the higher deformation theory of the prestack A. This answers the open problem about the higher structure on the Gerstenhaber-Schack complex at once in the general prestack case. (C) 2018 Elsevier Inc. All rights reserved
Gerstenhaber structure on Hochschild cohomology of the Fomin–Kirillov algebra on 3 generators
No full textInternational audienceThe goal of this article is to compute the Gerstenhaber bracket of the Hochschild cohomology of the Fomin–Kirillov algebra on three generators over a field of characteristic different from 2 and 3. This is in part based on a general method we introduce to easily compute the Gerstenhaber bracket between elements of HH0(A) and elements of HHn(A) for n∈N0, the method by M. Suárez-Álvarez [J. Pure Appl. Algebra 221, No. 8, 1981–1998 (2017; Zbl 1392.16009)] to calculate the Gerstenhaber bracket between elements of HH1(A) and elements of HHn (A) for any n∈N0, as well as an elementary result that allows to compute the remaining brackets from the previous ones. We also show that the Gerstenhaber bracket of HH (A) is not induced by any Batalin–Vilkovisky generator
Alternate Approaches to the Cup Product and Gerstenhaber Bracket on Hochschild Cohomology
No full textThesis (Ph.D.)--University of Washington, 2015The Hochschild cohomology of an algebra is a derived invariant of the algebra which admits both a graded ring structure (called the cup product) and a compatible graded Lie algebra structure (called the Gerstenhaber bracket). The Lie structure is particularly important as it provides a means of addressing the deformation theory of the algebra . In this thesis we produce some new methods for analyzing the cup product and Gerstenhaber bracket on Hochschild cohomology. For the cup product we produce a number of new, and rather fundamental, relations between the theories of twisting cochains and Hochschild cohomology. In the case of a Koszul algebra , our results imply that the Hochschild cohomology ring of is a subquotient of the tensor product algebra A\ox A^! of with its Koszul dual . We also investigate the Hochschild cohomology of smash product algebras . (Here is an algebra equipped with an action of a Hopf algebra .) In this setting, we produce new methods for computing both the cup product and Gerstenhaber bracket. For the Gerstenhaber bracket in particular, we show that there is an intermediate cohomology which is a braided commutative algebra in the category of Yetter-Drinfeld modules over , admits a braided anti-commutative bracket , and can be used to recover both the cup product and Gerstenhaber bracket on the standard Hochschild cohomology of
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