Computing the Hochschild Cohomology Groups of Some Families of Incidence Algebras

The purpose of this article is to present some computations of Hochschild cohomology groups of particular classes of incidence algebras using one-point extensions and one-point coextensions.     

l-遗传代数的Hochschild上同调群

设 $\Lambda$ 是域$k$上的有限维代数. 则 $\Lambda$的低阶 Hochschild上同调群在有限维代数的表示理论中扮演着重要的角色. 该文得到了 $l$ -遗传代数的一阶和二阶Hochschild 上同调群的维数方程.     

On the Hochschild Cohomology of Triangular Algebras

Abstract

In order to study the Hochschild cohomology of an n-triangular algebra 𝒯 _n , we construct a spectral sequence, whose terms are parametrized by the length of the trajectories of the quiver associated with 𝒯 _n , and which converges to the Hochschild cohomology of 𝒯 _n . We describe explicitly its components and its differentials which are sums of cup products. In case n = 3 we study some properties of the differential at level 2. We give some examples of use of the spectral sequence and recover formulas for the dimension of the cohomology groups of particular cases of triangular algebras.     

关于遗传代数上的例外序列自同态代数的Hochschild上同调与同调

令$A$是代数闭域$k$上的一个有限维结合代数, $\mod A$是有限维左$A$-模范畴,$X_1,X_2,\ldots,X_n$是$\mod A$中的完全例外序列,再令$E$是$X_1,X_2,\ldots,X_n$的自同态代数,我们在本文内,研究了$E$的总体维数,计算了$E$的Hochschild上同调群和同调群.     

Fibonacci代数的Hochschild上同调群

设Λ_d是Fibonacci代数,基于对Bardzell极小投射双模分解的细致分析,用组合的方法清晰地计算了Fibonacci代数Λ_d的各阶Hochschild上同调群的维数．     

Hochschild Cohomology Rings of Temperley-Lieb Algebras

The authors first construct an explicit minimal projective bimodule resolution(P, δ) of the Temperley-Lieb algebra A, and then apply it to calculate the Hochschild cohomology groups and the cup product of the Hochschild cohomology ring of A based on a comultiplicative map Δ:P → PAP. As a consequence, the authors determine the multiplicative structure of Hochschild cohomology rings of both Temperley-Lieb algebras and representation-finite q-Schur algebras under the cup product by giving an explicit presentation by generators and relations.     

Z_n分次代数的Hochschild上同调群

本文给出了Z_n分次代数A的Hochschild上同调群的定义,对低阶Hochschild上同调群进行了刻画.利用第一阶Hochschild上同调群给出了Z_n分次代数为分次可分代数的充要条件,证明了第二阶Hochschild上同调群的零次分支与A的Hochschild扩张之间的一一对应关系.     

量子外代数的Galois覆盖代数的Hochschild上同调

本文研究一类量子代数$\Lambda^n_q$的Hochschild上同调.量子代数$\Lambda^n_q$的极小投射双模分解被构造, $\Lambda^n_q$的各阶Hochschild上同调群的维数被清晰的给出.此外,对一些特殊的情况, $\Lambda^n_q$的上同调环也被清晰的刻画.     

Artin Algebras with Loops but No Outer Derivations

We construct families of Artin algebras over fields of arbitrary characteristic that contain a loop in their ordinary quiver but admit no nontrivial outer derivation. This refuses the long-held belief that such algebras should not exist.     

Hochschild Cohomology of Quiver Algebras Defined by Two Cycles and a Quantum-Like Relation II

We consider quiver algebras A _q over a field k defined by two cycles and a quantum-like relation depending on a nonzero element q in k. We determine the Hochschild cohomology ring of A _q and give necessary and sufficient conditions for A _q to have the finitely generated Hochschild cohomology ring.     

Transfer in Hochschild Cohomology of Blocks of Finite Groups

We develop the notion of a cohomology ring of blocks of finite groups and study its basic properties by means of transfer maps between the Hochschild cohomology rings of symmetric algebras associated with bounded complexes of finitely generated bimodules which are projective on either side.     

对应于根双模的拟遗传代数的Hochschild上同调群

设A是有限表示型遗传代数A=kQ的投射模范畴proj A上的根双模rad(-,-)所对应的拟遗传代数,基于Bardzell构造的极小投射双模分解,A的各阶Hochschild上同调群的维数被清晰地计算．     

Hochschild Cohomology of Quiver Algebras Defined by Two Cycles and a Quantum-Like Relation

We consider quiver algebras A _q over a field k defined by two cycles and a quantum-like relation depending on a nonzero element q in k, and describe the minimal projective bimodule resolution of A _q . In particular, in the case q = 1, we determine the Hochschild cohomology ring of A ₁ and show that it is a finitely generated k-algebra. Moreover the Hochschild cohomology ring of A ₁ modulo nilpotence is isomorphic to the polynomial ring of two variables.     

Cohomology of oriented algebras

In this paper, our goal is to develop the equivariant version of Hochschild cohomology. In particular, we develop a cohomology theory for oriented algebras.     

A Comparison of Products in Hochschild Cohomology

We prove that varieties are uniquely determined by the derived category of 𝒪[[?]]-modules with coherent cohomology which is the same as 𝒪-modules proved by A. Bondal and D. Orlov. We also generalize a theorem of Orlov.     

Twisted Bimodules and Hochschild Cohomology for Self-injective Algebras of Class A n , II

Up to derived equivalence, the representation-finite self-injective algebras of class A _n are divided into the wreath-like algebras (containing all Brauer tree algebras) and the Möbius algebras. In Part I (Forum Math. 11 (1999), 177–201), the ring structure of Hochschild cohomology of wreath-like algebras was determined, the key observation being that kernels in a minimal bimodule resolution of the algebras are twisted bimodules. In this paper we prove that also for Möbius algebras certain kernels in a minimal bimodule resolution carry the structure of a twisted bimodule. As an application we obtain detailed information on subrings of the Hochschild cohomology rings of Möbius algebras.     

Batalin-Vilkovisky Algebras and Cyclic Cohomology of Hopf Algebras

We show that the Connes–Moscovici negative cyclic cohomology of a Hopf algebra equipped with a character has a Lie bracket of degree -2. More generally, we show that a cyclic operad with multiplication is a cocyclic module whose simplicial cohomology is a Batalin–Vilkovisky algebra and whose negative cyclic cohomology is a graded Lie algebra of degree -2. This generalizes the fact that the Hochschild cohomology algebra of a symmetric algebra is a Batalin–Vilkovisky algebra.     

一类量子Koszul代数的${\mathbb{Z}}_{2}$-Galois 覆盖的 Hochschild上同调

考虑一类量子Koszul代数的 ${\mathbb{Z}}_{2}$-Galois覆盖$\Lambda_{\q}$, 并计算 这类代数的各阶Hochschild上同调群的维数, 进而利用道路的语言, 刻画了 Hochschild上同调环的cup积. 作为应用, 给出了这类代数的Hochschild上同调环模掉幂零理想的 代数结构.     

一个Cluster-Tilted代数的Hochschild上同调环

基于Furuya构造的一个cluster-tilted代数的极小投射双模分解,定义了该投射分解的所谓"余乘"结构,从而证明了该代数的Hochschild上同调环的cup积本质上是平行路的毗连并由此得到了该代数的Hochschild上同调环的一个由生成元与关系给出的实现.     

Hochschild Cohomology of an Algebra Associated with a Circular Quiver

In this article we show that an algebra A = K Γ/(f(X ^s )) has a periodic projective bimodule resolution of period 2, where KΓ is the path algebra of the circular quiver Γ with s vertices and s arrows over a commutative ring K, f(x) is a monic polynomial over K and X is the sum of all arrows in KΓ. Moreover, by means of this projective bimodule resolution, we compute the Hochschild cohomology group of A, and we give a presentation of the Hochschild cohomology ring HH?(A) by the generators and the relations in the case K is a field.