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 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.     

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.

     

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.     

Gerstenhaber bracket on Hopf algebra and Hochschild cohomologies

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 cohomology.     

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.     

Bounded Hochschild cohomology of Banach algebras with a matrix-like structure

Let be a unital Banach algebra. A projection in which is equivalent to the identitity may give rise to a matrix-like structure on any two-sided ideal in . In this set-up we prove a theorem to the effect that the bounded cohomology vanishes for all . The hypotheses of this theorem involve (i) strong H-unitality of , (ii) a growth condition on diagonal matrices in , and (iii) an extension of in by an amenable Banach algebra. As a corollary we show that if is an infinite dimensional Banach space with the bounded approximation property, is an infinite dimensional -space, and is the Banach algebra of approximable operators on , then for all .

     

Derivations of generalized Weyl algebras

A class of the associative and Lie algebras A[D] = A F[D] of Weyl type are studied, where A is a commutative associative algebra with an identity element over a field F of characteristic zero, and F[D] is the polynomial algebra of a finite dimensional commutative subalgebra of locally finite derivations of A such that A is D-simple. The derivations of these associative and Lie algebras are precisely determined.     

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.     

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.     

Isomorphism problems and groups of automorphisms for generalized Weyl algebras

We present solutions to isomorphism problems for various generalized Weyl algebras, including deformations of type-A Kleinian singularities and the algebras similar to introduced by S. P. Smith. For the former, we generalize results of Dixmier on the first Weyl algebra and the minimal primitive factors of by finding sets of generators for the group of automorphisms.

     

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.