Benjamin Briggs, Sarah Witherspoon
For a tensor product of algebras twisted by a bicharacter, we completely describe its Hochschild cohomology, as a Gerstenhaber algebra, in terms of the Hochschild cohomology of its component parts. This description generalizes a result of Bergh and Oppermann. It allows us to significantly simplify various calculations in the literature, and to compute Hochschild cohomology for a number of new examples.
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We compute the Hochschild cohomology of any block of $q$-Schur algebras. We focus the even part of this Hochschild cohomology ring. To compute the Hochschild cohomology of $q$-Schur algebras, we prove the following two results: first, we construct two graded algebra surjections between the Hochschild cohomologies of quasi-hereditary algebras because all $q$-Schur algebras over a field are quasi-hereditary. Second, we give the graded algebra isomorphism of Hochschild cohomologies by using a certain derived equivalence.
We show how to compute the low Hochschild cohomology groups of a partial relation extension algebra.
Hochschild cohomology of truncated quiver algebras
对于任意特征域上的截断代数，计算了它的Hochschild上同调.进一步证明了它的Hochschild上同调代数是有限维的当且仅当它的整体维数是有限的当且仅当它的子空间没有定向圈。
Hochschild cohomology via incidence algebras
给定一个代数A，我们联系一个关联代数A（A），并比较它们的Hochschild上同调群.
The Hochschild cohomology ring of a class of special biserial algebras
考虑域$K$上的一类特殊自内射双同调代数$Λ_N$，证明了$Λ_N$的Hochschild上同调环是一个自生成的$K$ -代数.而且$Λ_N$模零的Hochschild上同调环是一个克鲁尔维数2的可交换$K$ -代数.作为一个后果的猜想Snashall-Solberg \cite{SS}，关于Hochschild上同调环模零，持有这类代数。
We study cup product and cap product in Tate-Hochschild theory for a finite dimensional Frobenius algebra. We show that Tate-Hochschild cohomology ring equipped with cup product is isomorphic to singular Hochschild cohomology ring introduced by Wang. An application of cap product occurs in Tate-Hochschild duality; as in Tate (co)homology of a finite group, the cap product with the fundamental class of a finite dimensional Frobenius algebra provides certain duality result between Tate-Hochschild cohomology and homology groups. Moreover, we characterize minimal complete resolutions over a finite…
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 objective of this paper is to give a concrete interpretation of the dimension of the first Hochschild cohomology space of a cyclically oriented or tame cluster tilted algebra in terms of a numerical invariant arising from the potential.
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。
The first Hochschild cohomology group of a schurian cluster-tilted algebra
给定一个簇倾斜代数B，我们研究它的系数在B-B-双模B中的第一Hochschild上同调群HH ^[1]（B）。我们发现几个后果时，B是代表有限的，也在B是集群倾斜的类型\tilde{\mathbb{A}}的情况下。
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。
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.
The Lie algebra structure on the first Hochschild cohomology group of a monomial algebra
本文研究了有限维单项式代数A的第一Hochschild上同调群的李代数结构，利用它的群的组合学，在任意特征下.这使我们也可以检查的单位连通区的代数群的外部自同构的A在特征零。给出了（半）单纯性、可解性、可约性、可交换性和幂零性的判别准则.
For algebras with radical square zero, we give a combinatorial description to the Tate-Hochschild cohomology. We compute the Gerstenhaber algebra structure on the Tate-Hochschild cohomology for some classes of such algebras.
Gerstenhaber brackets on Hochschild cohomology of twisted tensor products
我们构造了一个扭曲的张量积的代数的Hochschild上同调的Gerstenhaber括号，并作为例子，计算Gerstenhaber括号的Buchweitz，Green，Madsen和Solberg的工作中产生的一些量子完全相交。证明了扭曲张量积的Hochschild上同调环的一个子代数（其上扭曲是平凡的）与因子的Hochschild上同调环的各个子代数的张量积同构，就像Gerstenhaber代数一样。
Tate and Tate-Hochschild Cohomology for finite dimensional Hopf Algebras
设A是域k上的任意有限维霍普夫代数。我们定义了A的Tate和Tate-Hochschild上同调，并引入了使A成为分次环的上积.我们建立了这些环之间的关系。特别地，A的Tate-Hochschild上同调同构（作为代数）于其系数在伴随模中的Tate上同调。因此，A的Tate上同调环是它的Tate-Hochschild上同调环的直和项。作为一个例子，我们显式地计算了Sweedler代数H_4的Tate和Tate-Hochschild上同调。
On the Hochschild Cohomology for Frobenius Kernels
本文利用谱序列方法研究Frobenius核的Hochschild上同调，并完整描述了SL₂的第一个Frobenius核G₁的Hochschild上同调G-代数结构。
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.
In this note we give a generalization for the higher order Hochschild cohomology and show that the secondary Hochschild cohomology is a particular case of this new construction.
Operations on the Secondary Hochschild Cohomology
我们表明，二级Hochschild上同调与三重$(A,B,\varepsilon)$有几个通常的Hochschild上同调的性质。证明了杯积和Lie积的存在性，讨论了$B$ -代数与扩张的关系，给出了次Hochschild上同调的Hodge型分解.
It has been shown in previous work that the modular group acts projectively on the center of a factorizable ribbon Hopf algebra. The center is the zeroth Hochschild cohomology group. In this article, we extend this projective action of the modular group to an arbitrary Hochschild cohomology group of a factorizable ribbon Hopf algebra, in fact up to homotopy even to a projective action on the entire Hochschild cochain complex.
Keller proved in 1999 that the Gerstenhaber algebra structure on the Hochschild cohomology of an algebra is an invariant of the derived category. In this paper, we adapt his approach to show that the Gerstenhaber algebra structure on the Tate-Hochschild cohomology of an algebra is preserved under singular equivalences of Morita type with level, a notion introduced by the author in previous work.
The Tate-Hochschild cohomology ring of a group algebra
证明了有限群代数$RG$的Tate-Hochschild上同调环$HH^*(RG,RG)$同构于$G$的共轭类代表的中心化子的Tate上同调环的直和.此外，我们的主要结果提供了一个明确的公式，上积在$HH^*(RG,RG)$关于这个分解。作为一个例子，这个公式帮助我们计算对称群$S_3$的系数在特征为3的域中的Tate-Hochschild上同调环。
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…
Hochschild cohomology of differential operators in positive characteristic
本文在正特征域的光滑簇上计算了Grothendieck微分算子的Hochschild上同调，并表明其结果由Frobenius作用于结构层上同调的导出逆极限给出。
The Lie structure on the Hochschild cohomology of a modular group algebra
证明了正特征域上循环群的群代数的Hochschild上同调的Gerstenhaber括号不是平凡的.在这种情况下，我们联系的李代数结构的奇数次的Hochschild上同调与一个Witt型代数。
We present a deformation theory associated to the higher Hochschild cohomology $H_{S^2}^*(A,A)$. We also study a $G$-algebra structure associated to this deformation theory.
Secondary Hochschild Cohomology
本文对$B$ -代数$A$定义了一个新的上同调理论.我们使用这个上同调来研究具有$B$ -代数结构的代数$A[[t]]$的变形。
Generalized parallel paths method for computing the first Hochschild cohomology group with applications to Brauer graph algebras
本文提出一种广义并行路径方法用于计算第一Hochschild上同调群，并应用于Brauer图代数及其关联代数的李结构比较。
Dimer models provide a method of constructing noncommutative crepant resolutions of affine toric Gorenstein threefolds. In homological mirror symmetry, they can also be used to describe noncommutative Landau--Ginzburg models dual to punctured Riemann surfaces. For a zigzag consistent dimer embedded in a torus, we explicitly compute the Hochschild cohomology of its Jacobi algebra in terms of dimer combinatorics. This includes a full characterization of the Batalin--Vilkovisky structure induced by the Calabi--Yau structure of the Jacobi algebra. We then compute the compactly supported Hochschild…
A family of Koszul self-injective algebras with finite Hochschild cohomology
本文给出了一类无限族的Koszul自内射代数，其Hochschild上同调环是有限维的.此外，对于每个$N \geq 5$，我们给出了一个例子，其中Hochschild上同调环的维数为$N$。这个代数家族包括并推广了Buchweitz、Green、Madsen和Solberg的四维Koszul自内射局部代数，它们被用来对Happel的问题给出否定的答案，因为它们具有无限的整体维数但有限维Hochschild上同调。
We identify two Batalin-Vilkovisky algebra structures, one obtained by Kowalzig and Krahmer on the Hochschild cohomology of an Artin-Schelter regular algebra with semisimple Nakayama automorphism and the other obtained by Lambre, Zhou and Zimmermann on the Hochschild cohomology of a Frobenius algebra also with semisimple Nakayama automorphism, provided that these two algebras are Koszul dual to each other.
Let $G$ be a finite simple group and $k$ be an algebraically closed field of prime characteristic dividing the order of $G$. We show that for all $2$-cocycles $α\in Z^2(G;k^\times)$, the first Hochschild cohomology group of the twisted group algebra $HH^1(k_αG)$ is nonzero.
Hochschild cohomology of fully group-graded algebras as Mackey functor
证明了一类全群分次分次代数的Hochschild上同调是Mackey函子。我们利用对称代数的Hochschild上同调之间的转移映射机制。
Symmetric cohomology and symmetric Hochschild cohomology of cocommutative Hopf algebras
本文将群的对称上同调推广到余交换Hopf代数，建立对称上同调与对称Hochschild上同调的同构，并研究其与经典上同调一致的条件。
Let $Γ$ be a connected graph without loops, cycles or multiple edges and $Z(Γ)$ the corresponding zigzag algebra. Then every Jordan derivation of $Z(Γ)$ is a derivation. Moreover, we will prove that the dimension of 1th Hochschild cohomology group of $Z(Γ)$ is one by computing the dimensions of linear spaces spanned by derivations and inner derivations. This implies that the dimension of the 1th Hochschild cohomology group of each algebra derived equivalent to a zigzag algebra is 1.
Hochschild cohomology of triangular matrix algebras
证明了三角形矩阵代数的Hochschild上同调与其分支子代数的霍赫希尔德同调之间存在两个长正合序列.我们还研究了第一序列的映射的结构。
The second Hochschild cohomology and deformations of Brauer graph algebras
本文研究了双分Brauer图代数的第二Hochschild上同调群，并通过表面模型解释了某些标准余循环对应的变形几何意义。
Hochschild cohomology for a class of some self-injective special biserial algebras of rank four
本文构造了Grothendieck群秩为$4$的特殊自内射双模代数$A_{T}$（$T\geq0$）的显式极小投射双模分解.作为主要结果，我们完全确定了$i\geq0$的PDMATHBMM的Hochschild上同调群${\rm HH}^{i}(A_{T})$的维数.此外，在$T=0$的情形下，利用生成元和关系给出了$A_{T}$的模零元${\rm HH}^{*}(A_{T})/\mathcal{N}_{A_{T}}$的Hochschild上同调环.
In this paper we define a new cohomology for multiplicative Hom-associative algebras, which generalize Hochschild cohomology and fits with deformations of Hom-associative algebras including the structure map $α$. It is a generalization of the known Hochschild-type cohomology for Hom-associative algebras defined by Amar, Ejbehi and Makhlouf. Moreover, we provide various observations and similarly a new cohomology of Hom-bialgebras extending the Gerstenhaber-Schack cohomology for Hom-bialgebras given by Dekkar and Makhlouf.
We describe the Gerstenhaber algebra structure on the Hochschild cohomology HH*$(A)$ when $A$ is a quadratic string algebra. First we compute the Hochschild cohomology groups using Barzdell's resolution and we describe generators of these groups. Then we construct comparison morphisms between the bar resolution and Bardzell's resolution in order to get formulae for the cup product and the Lie bracket. We find conditions on the bound quiver associated to string algebras in order to get non-trivial structures.
Hochschild cohomology in toposes
本文在环化拓扑斯中建立内部Hochschild上同调理论，构造内部Hochschild上链复形，特化至差分集合拓扑斯得到内部差分Hochschild上同调，并通过谱序列与绝对上同调比较，系统处理张量积以完善文献。
Hochschild Cohomology and Deformations of Clifford-Weyl Algebras
本文从玻色-费米统计量出发，对Clifford-Weyl代数${\mathcal C}(n,2k)$进行了全面的研究，包括Hochschild上同调（系数为自身）.我们证明了当$n$为偶数或当$k \neq 1$为偶数时，${\mathcal C}(n,2k)$为刚性.我们发现所有的非平凡变形的${\mathcal C}(2n+1,2)$和研究他们的表示。
Let k be a field, q in k. We derive a cup product formula on the Hochschild cohomology ring of a family Lambda_q of quiver algebras. Using this formula, we determine a subalgebra of k[x,y] isomorphic to Hochschild cohomology modulo N, where N is the ideal generated by homogeneous nilpotent elements. We explicitly construct non-nilpotent Hochschild cocycles which cannot be generated by lower homological degree elements, thus disproving the Snashall-Solberg finite generation conjecture.
Hochschild cohomology of the Weyl conformal algebra with coefficients in finite modules
本文通过将代数离散Morse理论推广到微分代数，完整计算了Weyl结合共形代数在所有有限模中系数的Hochschild上同调群。
We apply discrete algebraic Morse theory to calculate the Anick resolution of the group algebra of the group $G_3^2$. As a corollary, we evaluate Hochschild cohomologies of $G_3^2$ with coefficients in all 1-dimensional bimodules. Almost all these groups are trivial, the only exceptions are 1-dimensional $H^2$ for two particular 1-dimensional bimodules.
The first Hochschild cohomology group of quantum matrices and the quantum special linear group
在一般情况下，当形变参数不是单位根时，计算了量子矩阵的第一Hochschild上同调群、量子一般线性群和量子特殊线性群.作为推论，我们得到了这些代数的扭曲霍赫希尔德同调的信息。
Singular Hochschild Cohomology and Gerstenhaber Algebra Structure
本文将结合$k$ -代数$A$的奇异Hochschild上同调群$HH_{sg}^i(A, A)$定义为$i\in \mathbb{Z}$的奇异范畴$D_{sg}(A\otimes_k A^{op})$中从$A$到$A[i]$的态射.证明了$HH_{sg}^*(A, A)$具有Gerstenhaber代数结构，并且在对称代数$A$的情况下，$HH_{sg}^*(A, A)$是Batalin-Vilkovisky（BV）代数.
Hochschild cohomology of smash products and rank one Hopf algebras
给出了代数与霍普夫代数的Smash积的Hochschild上同调的环结构的一些一般性结果。我们计算这个环结构明确的一个大类的有限维霍普夫代数秩一。
In this paper, we compute the dimension of the Hochschild cohomology groups of any $m$-cluster tilted algebra of type $\tilde{\mathbb{A}}$. Moreover, we give conditions on the bounded quiver of an $m$-cluster tilted algebra $Λ$ of type $\tilde{\mathbb{A}}$ such that the Gerstenhaber algebra $\operatorname{HH}^*(Λ)$ has non-trivial multiplicative structures. We also show that the derived class of gentle $m$-cluster tilted algebras is not always completely determined by the Hochschild cohomology.
We study a Batalin-Vilkovisky algebra structure on the Hochschild cohomology of the group ring of finitely generated abelian groups. The Batalin-Vilkovisky algebra structure for finite abelian groups comes from the fact that the group ring of finite groups is a symmetric algebra, and the Batalin-Vilkovisky algebra structure for free abelian groups of finite rank comes from the fact that its group ring is a Calabi-Yau algebra.
Hochschild-Mitchell cohomology and Galois extensions
我们定义了$H$ -Galois扩张的$k$ -线性范畴和一个霍普夫代数$H$和证明了存在的Grothendieck谱序列的Hochschild-Mitchell上同调，与这种情况有关。这个谱序列是乘法的，对于一个$H$，一个群代数分解为一个由群的共轭类的集合索引的直和。我们还计算了具有无限个伴随群的范畴的Hochschild-Mitchell上同调群。
Hochschild cohomology groups of 5-dimensional complex nilpotent associative algebras
本文研究了满足特定条件的5维复Nilpotent结合代数的Hochschild上同调群，并通过显式矩阵形式展示了其计算结果，揭示了上同调在代数分类中的作用。
We compute the Hochschild cohomology ring of the algebras $A= k\langle X, Y\rangle/ (X^a, XY-qYX, Y^a)$ over a field $k$ where $a\geq 2$ and where $q\in k$ is a primitive $a$-th root of unity. We find the the dimension of $\mathrm{HH}^n(A)$ and show that it is independent of $a$. We compute explicitly the ring structure of the even part of the Hochschild cohomology modulo homogeneous nilpotent elements.
The action of the Nakayama automorphism of a Frobenius algebra on Hochschild cohomology
本文证明了Frobenius代数的中山自同构在Hochschild上同调上作用平凡，并据此构造了与代数、其自同构及导子相关的不变量。
BV structure on the Hochschild cohomology of twisted tensor products
本文用直和项上的扭曲 BV 算子描述了被双特征标扭曲的张量积的 Hochschild 上同调上的 BV 算子，推广了已有结果并简化了文献计算。
On the Hochschild cohomology ring of the quaternion group of order eight in characteristic two
设$k$是特征为2的代数闭域，$Q_8$是$8$阶四元数群.确定了群代数$kQ_8$的Hochschild上同调环上的Gerstenhaber李代数结构和Batalin-Vilkovisky结构.
This document is supplementary material for the author's paper \emph{The cup product on hochschild cohomology for localizations of filtered koszul algebras}, arXiv:1304.0527. We give a detailed computation of the Hochschild cohomology algebra for the Down-Up algebra with parameters $(0,1,0)$.
Let $a$ and $b$ be two coprime positive integers and $k$ an arbitrary field. We determine the ring structure of the Hochschild cohomology of the numerical semigroup algebras $k[s^{a},s^{b}]$ of embedding dimension two (thus also complete intersections) in terms of generators and relations. In addition, we compute the Hilbert series for this cohomology ring.
Dyer-Lashof-Cohen operations in Hochschild cohomology
我们给出了Hochschild上同调中类似于双闭路空间中同调的运算的显式公式。作为推论，我们得到了任何有限特征的支撑代数都是限制李代数。
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