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.     

Hochschild cohomology of a strongly homotopy commutative algebra

The Hochschild cohomology of a DG algebra A with coefficients in itself is, up to a suspension of degrees, a graded Lie algebra. The purpose of this paper is to prove that a certain DG Lie algebra of derivations appears as a finite codimensional graded sub Lie algebra of this Lie algebra when A is a strongly homotopy commutative algebra whose homology is concentrated in finitely many degrees. This result has interesting implications for the free the loop space homology which we explore here as well.     

The first Hochschild cohomology group of a schurian cluster-tilted algebra

Given a cluster-tilted algebra B we study its first Hochschild cohomology group HH¹(B) with coefficients in the B-B-bimodule B. We find several consequences when B is representation-finite, and also in the case where B is cluster-tilted of type . M. J. Redondo is a researcher from CONICET, Argentina.     

The cohomology ring of a monomial algebra

Summary In this paper we study the algebra structure of the cohomology ring of a monomial algebra. This article was processed by the author using the IAT_EX style filecljour1 from Springer-Verlag.     

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.     

The Lie structure on the Hochschild cohomology of a modular group algebra

We prove that the Gerstenhaber bracket on the Hochschild cohomology of the group algebra of a cyclic group over a field of positive characteristic is not trivial. In this case, we relate the Lie algebra structure on the odd degrees of the Hochschild cohomology with a Witt-type algebra.     

On the first Hochschild cohomology group of an algebra

Let A≡KΔ /I be a factor of a path algebra. We develop a strategy to compute dim H ¹(A), the dimension of the first Hochschild cohomology group of A, using combinatorial data from (Δ,I). That allows us to connect dim H ¹(A) with the rank and p-rank of the fundamental group π₁(Δ,I) of (Δ,I). We get explicit formulae for dim H ¹(A), when every path in Δ parallel to an arrow belongs to I or when I is homogeneous. Received: 12 April 1999 / Revised version: 9 October 2000     

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.     

The Hochschild cohomology of a closed manifold

Let M be a closed orientable manifold of dimension d and be the usual cochain algebra on M with coefficients in a field k. The Hochschild cohomology of M, is a graded commutative and associative algebra. The augmentation map induces a morphism of algebras . In this paper we produce a chain model for the morphism I. We show that the kernel of I is a nilpotent ideal and that the image of I is contained in the center of , which is in general quite small. The algebra is expected to be isomorphic to the loop homology constructed by Chas and Sullivan. Thus our results would be translated in terms of string homology.     

Hochschild cohomology ring modulo nilpotence of a one point extension of a quiver algebra defined by two cycles and a quantum-like relation

We consider a one point extension algebra B of a quiver algebra 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 B modulo nilpotence and show that if q is a root of unity, then B is a counterexample to Snashall-Solberg’s conjecture.     

On the Hochschild cohomology ring of tensor products of algebras

We prove that, as Gerstenhaber algebras, the Hochschild cohomology ring of the tensor product of two algebras is isomorphic to the tensor product of the respective Hochschild cohomology rings of these two algebras, when at least one of them is finite dimensional. In case of finite dimensional symmetric algebras, this isomorphism is an isomorphism of Batalin–Vilkovisky algebras. As an application, we explain by examples how to compute the Batalin–Vilkovisky structure, in particular, the Gerstenhaber Lie bracket, over the Hochschild cohomology ring of the group algebra of a finite abelian group.     

Hochschild cohomology of triangular string algebras and its ring structure

We compute the Hochschild cohomology groups HH^?(A)${\mathrm{HH}}^{?}(A)$ in case A is a triangular string algebra, and show that its ring structure is trivial.     

A family of Koszul self-injective algebras with finite Hochschild cohomology

This paper presents an infinite family of Koszul self-injective algebras whose Hochschild cohomology ring is finite-dimensional. Moreover, for each $N?5$ we give an example where the Hochschild cohomology ring has dimension $N$. This family of algebras includes and generalizes the 4-dimensional Koszul self-injective local algebras of [R.-O. Buchweitz, E.L. Green, D. Madsen, Ø. Solberg, Finite Hochschild cohomology without finite global dimension, Math. Res. Lett. 12 (2005) 805–816] which were used to give a negative answer to Happel’s question, in that they have infinite global dimension but finite-dimensional Hochschild cohomology.     

The hochschild cohomology ring of a modular group algebra: the commutative case

Let k be a field of characteristic P >0 and let G be a finite abelian group. We determine the structure of the Hochschild cohomology ring of the group algebra k G. Moreover, we prove that for any finite group G the Krull dimension of H H ^*(k G) equals the p-rank of G.