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1.
We show that the Hochschild cohomology of a monomial algebra over a field of characteristic zero vanishes from degree two if the first Hochschild cohomology is semisimple as a Lie algebra. We also prove that first Hochschild cohomology of a radical square zero algebra is reductive as a Lie algebra. In the case of the multiple loops quiver, we obtain the Lie algebra of square matrices of size equal to the number of loops.  相似文献   

2.
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.  相似文献   

3.

We study the simple connectedness of the class of finite-dimensional algebras over an algebraically closed field for which the Auslander–Reiten quiver admits a separating family of almost cyclic coherent components. We show that a tame algebra in this class is simply connected if and only if its first Hochschild cohomology space vanishes.

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4.
Ayako Itaba 《代数通讯》2013,41(1):404-415
We calculate the dimensions of the Hochschild cohomology groups of a self-injective special biserial algebra Λ s obtained by a circular quiver with double arrows. Moreover, we give a presentation of the Hochschild cohomology ring modulo nilpotence of Λ s by generators and relations. This result shows that the Hochschild cohomology ring modulo nilpotence of Λ s is finitely generated as an algebra.  相似文献   

5.
Patrick Le Meur 《代数通讯》2013,41(4):1325-1340
Let A be a basic connected finite dimensional algebra over an algebraically closed field, with ordinary quiver without oriented cycles. Given a presentation of A by quiver and admissible relations, Assem and de la Peña have constructed an embedding of the space of additive characters of the fundamental group of the presentation into the first Hochschild cohomology group of A. We compare the embeddings given by the different presentations of A. In some situations, we characterise the images of these embeddings in terms of (maximal) diagonalizable subalgebras of the first Hochschild cohomology group (endowed with its Lie algebra structure).  相似文献   

6.
In this paper we find necessary and sufficient conditions for an al­gebra to be a monomial algebra. These are conditions on finite abelian group gradings of the algebra and the first Hochschild cohomology group of the associated covering algebra of the grading. In particular, for a certain class of algebras, we show that an algebra Λ is a monomial algebra if and only if H 1(Λ,Λ) is the reduced Euler characteristic of the quiver of Λ. The proof of this uses the theory of noncommutat ive Gröbner bases.  相似文献   

7.
The aim of this paper is to investigate the first Hochschild cohomology of admissible algebras which can be regarded as a generalization of basic algebras.For this purpose,the authors study differential operators on an admissible algebra.Firstly,differential operators from a path algebra to its quotient algebra as an admissible algebra are discussed.Based on this discussion,the first cohomology with admissible algebras as coefficient modules is characterized,including their dimension formula.Besides,for planar quivers,the fc-linear bases of the first cohomology of acyclic complete monomial algebras and acyclic truncated quiver algebras are constructed over the field fc of characteristic 0.  相似文献   

8.
The aim of this paper is to investigate the first Hochschild cohomology of admissible algebras which can be regarded as a generalization of basic algebras. For this purpose, the authors study differential operators on an admissible algebra. Firstly, differential operators from a path algebra to its quotient algebra as an admissible algebra are discussed. Based on this discussion, the first cohomology with admissible algebras as coefficient modules is characterized, including their dimension formula. Besides, for planar quivers, the $k$-linear bases of the first cohomology of acyclic complete monomial algebras and acyclic truncated quiver algebras are constructed over the field $k$ of characteristic $0$.  相似文献   

9.
10.
11.
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.  相似文献   

12.
13.
14.
We investigate the Galois coverings of weakly shod algebras. For a weakly shod algebra not quasi-tilted of canonical type, we establish a correspondence between its Galois coverings and the Galois coverings of its connecting component. As a consequence we show that a weakly shod algebra which is not quasi-tilted of canonical type is simply connected if and only if its first Hochschild cohomology group vanishes.  相似文献   

15.
Using Grothendieck's semicontinuity theorem for half-exact functors,we derive two semicontinuity results on Hochschild cohomology.We apply these to show that the first Hochschild cohomogy groupof the mesh algebra of a translation quiver over a domain vanishesif and only if the translation quiver is simply connected. Wethen establish an exact sequence relating the first Hochschildcohomology group of an algebra to that of the endomorphism algebraof a module and apply it to study the first Hochschild cohomologygroup of an Auslander algebra. Our main result shows that fora finite-dimensional and representation-finite algebra algebraA over an algebraically closed field with Auslander algebra the following conditions are equivalent:
  1. (1)A admits no outer derivation;
  2. (2) admits no outer derivations;
  3. (3) A is simply connected;
  4. (4) is strongly simply connected.
. 2000 Mathematics Subject Classification 16E30, 16G30.  相似文献   

16.
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.  相似文献   

17.
In this paper, we introduce and study a class of algebras which we call ada algebras. An artin algebra is ada if every indecomposable projective and every indecomposable injective module lies in the union of the left and the right parts of the module category. We describe the Auslander–Reiten components of an ada algebra which is not quasi-tilted, showing in particular that its representation theory is entirely contained in that of its left and right supports, which are both tilted algebras. Also, we prove that an ada algebra over an algebraically closed field is simply connected if and only if its first Hochschild cohomology group vanishes.  相似文献   

18.
In this paper, we study the covering theory of laura algebras. We prove that if a connected laura algebra is standard (that is, has a standard connecting component), then it has Galois coverings associated to the coverings of the connecting component. As a consequence, the first Hochschild cohomology group of a standard laura algebra vanishes if and only if it has no proper Galois coverings.  相似文献   

19.
《代数通讯》2013,41(10):4871-4897
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.  相似文献   

20.
We study the Lie algebra structure of the first Hochschild cohomology group of a finite dimensional monomial algebra Λ, in terms of the combinatorics of its quiver, in any characteristic. This allows us also to examine the identity component of the algebraic group of outer automorphisms of Λ in characteristic zero. Criteria for the solvability, the (semi-) simplicity, the commutativity and the nilpotency are given. To cite this article: C. Strametz, C. R. Acad. Sci. Paris, Ser. I 334 (2002) 733–738.  相似文献   

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