Juan Carlos Bustamante, Patrick LeMeur
We establish a link between the Gerstenhaber bracket in the Hochshild cohomology and the behaviour of cycles in the module category of a monomial quadratic algebra $A$.
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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 cumber…
We introduce cyclic bilinear forms on coalgebras and use them to generalize Van den Bergh's Poisson brackets in representation algebras.
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.
Gerstenhaber brackets on Hochschild cohomology of general twisted tensor products
本文提出了利用扭曲张量积分解和任意双模分解上的最新结果，计算一般扭曲张量积代数的Hochschild上同调上Gerstenhaber括号的技术。
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.
We construct chain maps between the bar and Koszul resolutions for a quantum symmetric algebra (skew polynomial ring). This construction uses a recursive technique involving explicit formulae for contracting homotopies. We use these chain maps to compute the Gerstenhaber bracket, obtaining a quantum version of the Schouten-Nijenhuis bracket on a symmetric algebra (polynomial ring). We compute brackets also in some cases for skew group algebras arising as group extensions of quantum symmetric algebras.
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 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 Gerstenhaber algebras, to the tensor product of the respective subalgebras of the Hochschild cohomology rings of the factors.
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. 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 co…
We prove formulas of different types that allow 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. Also we give some new formulas for the Connes' differential on the Hochschild homology that lead to formulas for BV differential on the Hochschild cohomology in the case of symmetric algebras. Finally, we use one of the obtained formulas to get a full description of…
We define Gerstenhaber's graded Lie bracket directly on complexes other than the bar complex, under some conditions. The Koszul complex of a Koszul algebra in particular satisfies our conditions. As examples we recover the Schouten-Nijenhuis bracket for a polynomial ring and the Gerstenhaber bracket for a group algebra of a cyclic group of prime order.
In this monograph, we extend S. Schwede's exact sequence interpretation of the Gerstenhaber bracket in Hochschild cohomology to certain exact and monoidal categories. Therefore we establish 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, we show that our construction behaves well with respect to structure preserving functors between exact monoidal categories…
We study the Lie module structure given by the Gerstenhaber bracket on the Hochschild cohomology groups of a monomial algebra with radical square zero. The description of such Lie module structure will be given in terms of the combinatorics of the quiver. The Lie module structure will be related to the classification of finite dimensional modules over simple Lie algebras when the quiver is given by the two loops and the ground field is the complex numbers.
We define a cup product on the Hochschild cohomology of an associative conformal algebra $A$, and show the cup product is graded commutative. We define a graded Lie bracket with the degree $-1$ on the Hochschild cohomology $\HH^{\ast}(A)$ of an associative conformal algebra $A$, and show that the Lie bracket together with the cup product is a Gerstenhaber algebra on the Hochschild cohomology of an associative conformal algebra. Moreover, we consider the Hochschild cohomology of split extension conformal algebra $A\hat{\oplus}M$ of $A$ with a conformal bimodule $M$, and show that there exist an…
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 mono…
Hochschild cohomology of graded skew-gentle algebras: Gerstenhaber algebra structure and geometric interpretation
本文研究了带权斜 gentle 代数的 Hochschild 上同调结构及其几何意义，揭示了其与底层带权表面的关系。
Homotopy liftings and Hochschild cohomology of some twisted tensor products
该研究利用Volkov的同伦提升方法，给出代数张量积及双特征扭曲张量积的Hochschild上同调同构结论的新证明，展现了同伦提升的理论应用价值。
Hochschild cohomology governs deformations of algebras, and its graded Lie structure plays a vital role. We study this structure for the Hochschild cohomology of the skew group algebra formed by a finite group acting on an algebra by automorphisms. We examine the Gerstenhaber bracket with a view toward deformations and developing bracket formulas. We then focus on the linear group actions and polynomial algebras that arise in orbifold theory and representation theory; deformations in this context include graded Hecke algebras and symplectic reflection algebras. We give some general results des…
The purpose of this paper is to define cohomology structures on Hom-associative algebras and Hom-Lie algebras. The first and second coboundary maps were introduced by Makhlouf and Silvestrov in the study of one-parameter formal deformations theory. Among the relevant formulas for a generalization of Hochschild cohomology for Hom-associative algebras and a Chevalley-Eilenberg cohomology for Hom-Lie algebras, we define Gerstenhaber bracket on the space of multilinear mappings of Hom-associative algebras and Nijenhuis-Richardson bracket on the space of multilinear mappings of Hom-Lie algebras. Al…
We formulate the Gerstenhaber algebra structure of Hochschild cohomology of finite group extensions of some quantum complete intersections. When the group is trivial, this work characterizes the graded Lie brackets on Hochschild cohomology of these quantum complete intersections, previously only known for a few cases. As an example, we compute the algebra structure for two generator quantum complete intersections extended by select groups.
We show that recollements of module categories give rise to homomorphisms between the associated Hochschild cohomology algebras which preserve the strict Gerstenhaber structure, i.e., the cup product, the graded Lie bracket and the squaring map. We review various long exact sequences in Hochschild cohomology and apply our results in order to realise that the occurring maps preserve the strict Gerstenhaber structure as well. As a byproduct, we generalise a known long exact cohomology sequence of Koenig-Nagase to arbitrary surjective homological epimorphisms. We use our observations to motivate …
In this paper, we study the Batalin-Vilkovisky structure on the Hochschild cohomology of quantum zigzag algebras $A_{q}$ of type $\widetilde{\mathbf{A}}_{1}$. We first calculate the dimensions of Hochschild homology groups and Hochschild cohomology groups of $A_{q}$. Based on these computations, we determine the Hochschild cohomology ring of $A_{q}$, and give the Batalin-Vilkovisky operator and the Gerstenhaber bracket on Hochschild cohomology ring of $A_{q}$ explicitly.
Hochschild cohomology of graded gentle algebras and intrinsic formality
本文研究了带状温和代数的Hochschild上同调及其内在形式，揭示了其与部分缠绕Fukaya范畴的关系，并提供了内在形式的特征化结果。
Graded Lie structure on cohomology of some exact monoidal categories
本文在正合幺半范畴中建立了李括号拓扑定义与代数定义的联系，证明了拓扑定义可产生Gerstenhaber代数结构，并将A∞-余导子与同伦提升技术推广到该范畴。
Structure constants for simple Lie algebras from principal $\mathfrak{sl}_2$-triple
本研究针对单复李代数，利用主sl₂-三元组和最高权向量诱导基，通过超越量计算slₙ在该基下的李括号，推广了sl₂的经典规则，采用经典不变量理论图形演算证明，并讨论了其他李代数类型。
The Hochschild cohomology and the Tamarkin-Tsygan calculus of gentle algebras
本文首次完整计算了温和代数族的Tamarkin-Tsygan演算，结合曲面几何模型解释了其与辛上同调的关联，并给出由该演算恢复温和代数结构结果的方法。
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 Gers…
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