Hochschild cohomology of special biserial algebras

Based on a four-term exact sequence,the formulae on the dimensions of the first and the second Hochschild cohomology groups of special biserial algebras with normed bases are obtained in terms of combinatorics.     

Koszul代数的Hochschild上同调环

基于Buchweitz等人对Koszul代数的Hochschild上同调环的乘法结构的细致分析,给出了Koszul代数的Hochschild上同调环的乘法本质上是平行路的毗连的一个充要条件,并由此重新证明了二次三角string代数的Hochschild上同调环的乘法是平凡的,从而改进了Bustamante的证明.     

On the simplicial cohomology of Banach operator algebras

We investigate the simplicial cohomology of certain Banach operator algebras. The two main examples considered are the Banach algebra of all bounded operators on a Banach space and its ideal of approximable operators. Sufficient conditions will be given forcing Banach algebras of this kind to be simplicially trivial.     

On Hochschild cohomology rings of Fibonacci algebras

Let Λ be a Fibonacci algebra over a field k. The multiplication of Hochschild cohomology ring of Λ induced by the Yoneda product is described explicitly. As a consequence, the multiplicative structure of Hochschild cohomology ring of Λ is proved to be trivial.     

Hochschild cohomology of truncated quiver algebras

For a truncated quiver algebra over a field of an arbitrary characteristic, its Hochschild cohomology is calculated. Moreover, it is shown that its Hochschild cohomology algebra is finitedimensional if and only if its global dimension is finite if and only if its quiver has no oriented cycles.     

一类量子Koszul 代数的Hochschild 上同调

本文利用组合的方法, 详细地计算了一类量子Koszul 代数Λ_q (q ∈ k \{0}) 的各阶Hochschild 上同调空间的维数, 清晰地刻划了代数Λ_q 的Hochschild 上同调的cup 积, 确定了代数Λ_q 的Hochschild上同调环HH^*(Λ_q) 模去幂零元生成的理想N 的结构, 证明了当q 为单位根时, HH^*(Λ_q)/N 作为代数不是有限生成的, 从而为Snashall-Solberg 猜想(即HH^*(Λ)/N 作为代数是有限生成的) 提供了更多反例.     

The Hochschild cohomology of the quasi-entwining structure

We give a concept of quasi-entwining structure,and investigate the Hochschild cohomology of the quasi-entwining structure.We obtain the equivalent theorems on the Hochschild cohomology of the quasientwining structure.In particular,we get the isomorphism theorem between the Hochschild cohomology of coalgebra structures and the Hochschild cohomology of the dual algebra structures for the quasi-entwining structures of finite-dimensional algebras and coalgebras.     

Hochschild cohomology and graded Hecke algebras

We develop and collect techniques for determining Hochschild cohomology of skew group algebras and apply our results to graded Hecke algebras. We discuss the explicit computation of certain types of invariants under centralizer subgroups, focusing on the infinite family of complex reflection groups to illustrate our ideas. Resulting formulas for Hochschild two-cocycles give information about deformations of and, in particular, about graded Hecke algebras. We expand the definition of a graded Hecke algebra to allow a nonfaithful action of on , and we show that there exist nontrivial graded Hecke algebras for , in contrast to the case of the natural reflection representation. We prove that one of these graded Hecke algebras is equivalent to an algebra that has appeared before in a different form.

     

Hochschild cohomology of ring objects in monoidal categories

We define the Hochschild complex and cohomology of a ring object in a monoidal category enriched over abelian groups. We interpret the cohomology groups and prove that the cohomology ring is graded-commutative.     

Hochschild cohomology of partial relation extensions

We show how to compute the low Hochschild cohomology groups of a partial relation extension algebra.     

Hochschild cohomology of truncated basic cycle

Dimensions of the Hochschild cohomology groups of a truncated algebra of a basic cycle are explicitly given. Project supported by the State Education Commission of China, the Chinese Academy of Sciences and the National Natural Science Foundation of China.     

The Hochschild cohomology of square-free monomial complete intersections

We determine the Hochschild cohomology algebras of the square-free monomial complete intersections. In particular we provide an explicit formula for the cup product which gives the cohomology module an algebra structure and then we describe this structure in terms of generators and relations. In addition, we compute the Hilbert series of the Hochschild cohomology of these algebras.     

The Hochschild cohomology ring of a cyclic block

Suppose is a block of a group algebra with cyclic defect group. We calculate the Hochschild cohomology ring of , giving a complete set of generators and relations. We then show that if is the principal block, the canonical map from to the Hochschild cohomology ring of induces an isomorphism modulo radicals.     

Graded Hochschild cohomology of a path algebra with oriented cycles

The aim of this paper is to characterize the first graded Hochschild cohomology of a hereditary algebra whose Gabriel quiver is admitted to have oriented cycles. The interesting conclusion we have obtained shows that the standard basis of the first graded Hochschild cohomology depends on the genus of a quiver as a topological object. In this paper, we overcome the limitation of the classical Hochschild cohomology for hereditary algebra where the Gabriel quiver is assumed to be acyclic. As preparation, we first investigate the graded differential operators on a path algebra and the associated graded Lie algebra.     

Batalin–Vilkovisky algebra structures on Hochschild cohomology of generalized Weyl algebras

We devote to the calculation of Batalin–Vilkovisky algebra structures on the Hochschild cohomology of skew Calabi–Yau generalized Weyl algebras. We first establish a Van den Bergh duality at the level of complex. Then based on the results of Solotar et al., we apply Kowalzig and Krähmer's method to the Hochschild homology of generalized Weyl algebras, and translate the homological information into cohomological one by virtue of the Van den Bergh duality, obtaining the desired Batalin–Vilkovisky algebra structures. Finally, we apply our results to quantum weighted projective lines and Podleś quantum spheres, and the Batalin–Vilkovisky algebra structures for them are described completely.     

G-structure on the cohomology of Hopf algebras

We prove that is a Gerstenhaber algebra, where is a Hopf algebra. In case is the Drinfeld double of a finite-dimensional Hopf algebra , our results imply the existence of a Gerstenhaber bracket on . This fact was conjectured by R. Taillefer. The method consists of identifying as a Gerstenhaber subalgebra of (the Hochschild cohomology of ).

     

The ordinary quivers of Hochschild extension algebras for self-injective Nakayama algebras

Let T be a Hochschild extension algebra of a finite dimensional algebra A over a field K by the standard duality A-bimodule Hom_K(A, K). In this paper, we determine the ordinary quiver of T if A is a self-injective Nakayama algebra by means of the ?-graded second Hochschild homology group HH₂(A) in the sense of Sköldberg.