On Hochschild Cohomology of a Self-injective Special Biserial Algebra Obtained by a Circular Quiver with Double Arrows

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.     

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.     

The fundamental groups of a triangular algebra

A finite dimensional algebra A (over an algebraically closed field) is called triangular if its ordinary quiver has no oriented cycles. To each presentation (Q I) of A is attached a fundamental group π₁(Q I), and A is called simply connected if π₁(Q I) is trivial for every presentation of A. In this paper, we provide tools for computations with the fundamental groups, as well as criteria for simple connectedness. We find relations between the fundamental groups of A and the first Hochschild cohomology H ¹ (A A).     

The first cohomology group of trivial extensions of special biserial algebras

Given a finite dimensional special biserial algebra A with normed basis we obtain the dimension formulae of the first Hochschild homology groups of A and the vector space Alt(DA). As a consequence, an explicit dimension formula of the first Hochschild cohomology group of trivial extension TA = A×DA in terms of the combinatorics of the quiver and relations is determined.     

Gerstenhaber Algebra Structure on the Hochschild Cohomology of Quadratic String Algebras

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.     

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.     

An invariant characterization of monomial algebras

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.     

Hochschild Cohomology and Representation-Finite Algebras

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.     

On the Semisimplicity of the Outer Derivations of Monomial Algebras

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.     

Singular Hochschild cohomology via the singularity category

We show that the singular Hochschild cohomology (= Tate–Hochschild cohomology) of an algebra A is isomorphic, as a graded algebra, to the Hochschild cohomology of the differential graded enhancement of the singularity category of A. The existence of such an isomorphism is suggested by recent work by Zhengfang Wang.     

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.     

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.     

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.     

An Application of a Theorem of Sheila Brenner for Hochschild Extension Algebras of a Truncated Quiver Algebra

Let A be a truncated quiver algebra over an algebraically closed field such that any oriented cycle in the ordinary quiver of A is zero in A. We give the number of the indecomposable direct summands of the middle term of an almost split sequence for a class of Hochschild extension algebras of A by the standard duality module D(A).

     

Hochschild cohomology and “smoothness” in monoidal categories

We introduce and investigate the properties of Hochschild cohomology of algebras in an abelian monoidal category M. We show that the second Hochschild cohomology group of an algebra in M classifies extensions of A up to an equivalence. We characterize algebras of Hochschild dimension 0 (separable algebras), and of Hochschild dimension ≤1 (formally smooth algebras). Several particular cases and applications are included in the last section of the paper.     

The Simple Connectedness of Tame Algebras with Separating Almost Cyclic Coherent Auslander–Reiten Components

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.

     

Automorphism groups of trivial extensions

Given a split basic finite dimensional algebra A over a field, we study the relationship between the groups of categorical automorphisms of A and its trivial extension A?D(A). Our results cover all triangular algebras and all 2-nilpotent algebras whose quiver has no nontrivial oriented cycle of length ?2. In this latter as well as in the hereditary case, we give structure theorem for CAut(A?D(A)) in terms of CAut(A). As a byproduct, we get the precise relationship between the first Hochschild cohomology groups of A and A?D(A).     

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.