Gerstenhaber Bracket Research Articles

Gerstenhaber bracket on Hopf algebra and Hochschild cohomologies

We calculate the Gerstenhaber bracket on Hopf algebra and Hochschild cohomologies of the Taft algebra Tp 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 Tp, as in all known quasi-triangular Hopf algebras. This example is the first known bracket computation for a nonquasi-triangular algebra. 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.

Lie brackets on Hopf algebra cohomology

By work of Farinati, Solberg, and Taillefer, it is known 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 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, most of which are nonquasi-triangular.

Gerstenhaber algebra of ℤ2 × ℤ2-Galois covering of quantum exterior algebra

Let [Formula: see text] be the quantum exterior algebras over a field [Formula: see text] with char [Formula: see text], [Formula: see text] be the [Formula: see text]-Galois coverings of [Formula: see text]. In this paper, a basis of each degree of Hochschild cohomology of [Formula: see text], the cup product and the Gerstenhaber bracket on Hochschild cohomology of [Formula: see text] are given.

Cyclic Gerstenhaber–Schack cohomology

We 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–Vilkovisky) 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 e_2 -algebra structure but rather by the resulting e_3 -algebra structure, which is expressed in terms of the cup product and B .

Gerstenhaber structure on Hochschild cohomology of the Fomin–Kirillov algebra on 3 generators

The 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 \operatorname{HH}^0(A) and elements of \operatorname{HH}^n(A) for n \in \mathbb{N}_0 , 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 \operatorname{HH}^1(A) and elements of \operatorname{HH}^n(A) for any n \in \mathbb{N}_0 , 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 \operatorname{HH}^{\bullet}(A) is not induced by any Batalin–Vilkovisky generator.

Cup product and Gerstenhaber bracket on Hochschild cohomology of a family of quiver algebras

We present a cup product formula on the Hochschild cohomology of a family of quiver algebras. We use the formula to determine the set of homogeneous non-nilpotent Hochschild cocycles and construct a canonical isomorphism between Hochschild cohomology modulo nilpotents and a subalgebra of that is not finitely generated. For some members of the family, we present the Gerstenhaber ideal of homogeneous nilpotent cocycles using homotopy lifting technique. We then determine their Hochschild cohomology modulo the weak Gerstenhaber ideal generated by nilpotent elements, thereby providing an answer to a question of Reiner Hermann.

Homotopy liftings and Hochschild cohomology of some twisted tensor products

The Hochschild cohomology of a tensor product of algebras is isomorphic to a graded tensor product of Hochschild cohomology algebras, as a Gerstenhaber algebra. A similar result holds when the tensor product is twisted by a bicharacter. We present new proofs of these isomorphisms, using Volkov’s homotopy liftings that were introduced for handling Gerstenhaber brackets expressed on arbitrary bimodule resolutions. Our results illustrate the utility of homotopy liftings for theoretical purposes.

GERSTENHABER BRACKETS FOR SKEW GROUP ALGEBRAS IN POSITIVE CHARACTERISTIC

The deformation theory of an algebra is controlled by the Gerstenhaber bracket, a Lie bracket on Hochschild cohomology. We develop techniques for evaluating Gerstenhaber brackets of semidirect product algebras recording actions of finite groups over fields of positive characteristic. The Hochschild cohomology and Gerstenhaber bracket of these skew group algebras can be complicated when the characteristic of the underlying field divides the group order. We show how to investigate Gerstenhaber brackets using twisted product resolutions, which are often smaller and more convenient than the cumbersome bar resolution typically used. These resolutions provide a concrete description of the Gerstenhaber bracket suitable for exploring questions in deformation theory. We demonstrate with the prototypical example of a Drinfeld Hecke algebra (graded Hecke algebra) in positive characteristic.

Gerstenhaber brackets on Hochschild cohomology of general twisted tensor products

We 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.

$A_{\infty}$-coderivations and the Gerstenhaber bracket on Hochschild cohomology

We show that Hochschild cohomology of an algebra over a field is a space of infinity coderivations on an arbitrary projective bimodule resolution of the algebra. The Gerstenhaber bracket is the graded commutator of infinity coderivations. We thus generalize, to an arbitrary resolution, Stasheff’s realization of the Gerstenhaber bracket on Hochschild cohomology as the graded commutator of coderivations on the tensor coalgebra of the algebra.

The Weil Algebra of a Double Lie Algebroid

Abstract Given a double vector bundle $D\to M$, we define a bigraded bundle of algebras $W(D)\to M$ called the “Weil algebra bundle”. The space ${\mathcal{W}}(D)$ of sections of this algebra bundle ”realizes” the algebra of functions on the supermanifold $D[1,1]$. We describe in detail the relations between the Weil algebra bundles of $D$ and those of the double vector bundles $D^{\prime},\ D^{\prime\prime}$ obtained from $D$ by duality operations. We show that ${\mathcal{V}\mathcal{B}}$-algebroid structures on $D$ are equivalent to horizontal or vertical differentials on two of the Weil algebras and a Gerstenhaber bracket on the 3rd. Furthermore, Mackenzie’s definition of a double Lie algebroid is equivalent to compatibilities between two such structures on any one of the three Weil algebras. In particular, we obtain a ”classical” version of Voronov’s result characterizing double Lie algebroid structures. In the case that $D=TA$ is the tangent prolongation of a Lie algebroid, we find that ${\mathcal{W}}(D)$ is the Weil algebra of the Lie algebroid, as defined by Mehta and Abad–Crainic. We show that the deformation complex of Lie algebroids, the theory of IM forms and IM multi-vector fields, and 2-term representations up to homotopy all have natural interpretations in terms of our Weil algebras.

Polysymplectic formulation for BF gravity with Immirzi parameter

The polysymplectic formulation of the CMPR action, which is a BF-type formulation of general relativity that involves an arbitrary Immirzi parameter, is performed. We implement a particular scheme within this covariant Hamiltonian approach to analyze the constraints that characterize the CMPR model. By means of the privileged -forms and the Poisson–Gerstenhaber bracket, inherent to the polysymplectic framework, the BF field equations associated to the CMPR action are obtained and, in consequence, the Einstein equations naturally emerge by solving the simplicity constraints of the theory. Further, from the polysymplectic analysis of the CMPR action the De Donder–Weyl Hamiltonian formulation of the Holst action is recovered, which is consistent with the Lagrangian analysis of this model as reported in the literature.

Singular Hochschild cohomology and algebraic string operations

Given a differential graded (dg) symmetric Frobenius algebra A we construct an unbounded complex \mathcal{D}^{*}(A,A) , called the Tate–Hochschild complex, which arises as a totalization of a double complex having Hochschild chains as negative columns and Hochschild cochains as non-negative columns. We prove that the complex \mathcal{D}^*(A,A) computes the singular Hochschild cohomology of A . We construct a cyclic (or Calabi–Yau) A -infinity algebra structure, which extends the classical Hochschild cup and cap products, and an L -infinity algebra structure extending the classical Gerstenhaber bracket, on \mathcal{D}^*(A,A) . Moreover, we prove that the cohomology algebra H^*(\mathcal{D}^*(A,A)) is a Batalin–Vilkovisky (BV) algebra with BV operator extending Connes' boundary operator. Finally, we show that if two dg algebras are quasi-isomorphic then their singular Hochschild cohomologies are isomorphic and we use this invariance result to relate the Tate–Hochschild complex to string topology.

Gerstenhaber Bracket on the Hochschild Cohomology via An Arbitrary Resolution

AbstractWe prove formulas of different types that allow us to calculate the Gerstenhaber bracket on the Hochschild cohomology of an algebra using some arbitrary projective bimodule resolution for it. Using one of these formulas, we give a new short proof of the derived invariance of the Gerstenhaber algebra structure on Hochschild cohomology. We also give some new formulas for the Connes differential on the Hochschild homology that lead to formulas for the Batalin–Vilkovisky (BV) differential on the Hochschild cohomology in the case of symmetric algebras. Finally, we use one of the obtained formulas to provide a full description of the BV structure and, correspondingly, the Gerstenhaber algebra structure on the Hochschild cohomology of a class of symmetric algebras.

The Cohomology Structure of Hom-H-Pseudoalgebras

The goal of this paper is to study cohomological theory of Hom-associative H-pseudoalgebras and Hom–Lie H-pseudoalgebras. We define Gerstenhaber bracket on the space of multilinear mappings of Hom-associative H-pseudoalgebra. Furthermore, the symmetric Schouten product and alternating Schouten product are studied. Using the Gerstenhaber bracket and alternating Schouten product, differential graded Lie algebra are constructed on the space of multilinear mappings of Hom-associative H-pseudoalgebra and Hom-Lie H-pseudoalgebras.

Brackets in representation algebras of Hopf algebras

For any graded bialgebras A and B , we define a commutative graded algebra A_B representing the functor of B -representations of A . When A is a cocommutative graded Hopf algebra and B is a commutative ungraded Hopf algebra, we introduce a method deriving a Gerstenhaber bracket in A_B from a Fox pairing in A and a balanced biderivation in B . Our construction is inspired by Van den Bergh’s non-commutative Poisson geometry, and may be viewed as an algebraic generalization of the Atiyah–Bott–Goldman Poisson structures on moduli spaces of representations of surface groups.

Moyal product for (n – 1)-forms within the covariant Hamiltonian formalism for fields

In these brief notes we analyze an associative product for (n – 1)- forms in the polymomentum phase space which is necessary in order to complete the deformation quantization scheme within the context of the multisymplectic formalism. In particular, we identify certain conditions that the proposed product must follow to generate a Moyal bracket induced from the Gerstenhaber bracket naturally emerging within this context.

Gerstenhaber brackets on Hochschild cohomology of twisted tensor products

We construct the Gerstenhaber bracket on Hochschild cohomology of a twisted tensor product of algebras, and, as examples, compute Gerstenhaber brackets for some quantum complete intersections arising in the work of Buchweitz, Green, Madsen, and Solberg.We prove that a subalgebra of the Hochschild cohomology ring of a twisted tensor product, on which the twisting is trivial, is isomorphic, as a Gerstenhaber algebra, to the tensor product of the respective subalgebras of the Hochschild cohomology rings of the factors.

De Donder–Weyl Hamiltonian formalism of MacDowell–Mansouri gravity

We analyse the behaviour of the MacDowell–Mansouri action with internal symmetry group under the De Donder–Weyl Hamiltonian formulation. The field equations, known in this formalism as the De Donder–Weyl equations, are obtained by means of the graded Poisson–Gerstenhaber bracket structure present within the De Donder–Weyl formulation. The decomposition of the internal algebra allows the symmetry breaking , which reduces the original action to the Palatini action without the topological term. We demonstrate that, in contrast to the Lagrangian approach, this symmetry breaking can be performed indistinctly in the polysymplectic formalism either before or after the variation of the De Donder–Weyl Hamiltonian has been done, recovering Einstein’s equations via the Poisson–Gerstenhaber bracket.

The Gerstenhaber bracket as a Schouten bracket for polynomial rings extended by finite groups

We apply new techniques to Gerstenhaber brackets on the Hochschild cohomology of a skew group algebra formed from a polynomial ring and a finite group (in characteristic 0). We show that the Gerstenhaber brackets can always be expressed in terms of Schouten brackets. We obtain as consequences some conditions under which brackets are always 0, strengthening known results.