Hochschild cohomology of m-cluster tilted algebras of type 𝔸̃

2020 ◽  
pp. 2150018
Author(s):  
Viviana Gubitosi
Keyword(s):  
Hochschild Cohomology ◽  
Cohomology Groups ◽  
Tilted Algebra ◽  
Type Formula ◽  
Tilted Algebras ◽  
Gerstenhaber Algebra ◽  
Cluster Tilted Algebra ◽  
Multiplicative Structures

In this paper, we compute the dimension of the Hochschild cohomology groups of any [Formula: see text]-cluster tilted algebra of type [Formula: see text]. Moreover, we give conditions on the bounded quiver of an [Formula: see text]-cluster tilted algebra [Formula: see text] of type [Formula: see text] such that the Gerstenhaber algebra [Formula: see text] has nontrivial multiplicative structures. We also show that the derived class of gentle [Formula: see text]-cluster tilted algebras is not always completely determined by the dimension of the Hochschild cohomology.

scholarly journals THE ALGEBRAS DERIVED EQUIVALENT TO GENTLE CLUSTER TILTED ALGEBRAS

2012 ◽  
Vol 11 (01) ◽  
pp. 1250012 ◽  
Author(s):  
GRZEGORZ BOBIŃSKI ◽  
ASLAK BAKKE BUAN
Keyword(s):  
Tilted Algebra ◽  
Type Formula ◽  
Finite Dimensional ◽  
Tilted Algebras ◽  
Dynkin Type ◽  
Euclidean Type ◽  
Finite Dimensional Algebras ◽  
Cluster Tilted Algebra

A cluster tilted algebra is known to be gentle if and only if it is cluster tilted of Dynkin type 𝔸 or Euclidean type [Formula: see text]. We classify all finite-dimensional algebras which are derived equivalent to gentle cluster tilted algebras.

scholarly journals THE FIRST HOCHSCHILD COHOMOLOGY GROUP OF A CLUSTER TILTED ALGEBRA REVISITED

2013 ◽  
Vol 23 (04) ◽  
pp. 729-744 ◽  
Author(s):  
IBRAHIM ASSEM ◽  
JUAN CARLOS BUSTAMANTE ◽  
KIYOSHI IGUSA ◽  
RALF SCHIFFLER
Keyword(s):  
Vector Space ◽  
Cohomology Group ◽  
Hochschild Cohomology ◽  
Tilted Algebra ◽  
Direct Sum ◽  
Hochschild Cohomology Group ◽  
Cluster Tilted Algebra

Given a cluster-tilted algebra B we study its first Hochschild cohomology group HH 1(B) with coefficients in the B–B-bimodule B. If C is a tilted algebra such that B is the relation extension of C by [Formula: see text], then we prove that HH 1(B) is isomorphic, as a vector space, to the direct sum of HH 1(C) with HH 1(B,E). This yields homological interpretations for results of the first and the fourth authors with M. J. Redondo.

scholarly journals τ-Tilting finite cluster-tilted algebras

2020 ◽  
Vol 63 (4) ◽  
pp. 950-955 ◽  
Author(s):  
Stephen Zito
Keyword(s):  
Tilted Algebra ◽  
Tilted Algebras ◽  
Cluster Tilted Algebra

We prove if B is a cluster-tilted algebra, then B is τB-tilting finite if and only if B is representation-finite.

Open Frobenius Cluster-Tilted Algebras

2022 ◽  
Vol 29 (01) ◽  
pp. 1-22
Author(s):  
Viviana Gubitosi
Keyword(s):  
Finite Type ◽  
Tilted Algebra ◽  
Frobenius Structures ◽  
Tilted Algebras ◽  
Cluster Tilted Algebra

In this paper, we compute the Frobenius dimension of any cluster-tilted algebra of finite type. Moreover, we give conditions on the bound quiver of a cluster-tilted algebra [Formula: see text] such that [Formula: see text] has non-trivial open Frobenius structures.

scholarly journals Comparison morphisms between two projective resolutions of monomial algebras

2017 ◽  
pp. 1-31
Author(s):  
María Julia Redondo ◽  
Lucrecia Román
Keyword(s):  
Lie Algebra ◽  
Hochschild Cohomology ◽  
Cohomology Groups ◽  
Efficient Tool ◽  
Simple Description ◽  
Monomial Algebra ◽  
Cup Product ◽  
Lie Bracket ◽  
Projective Resolutions ◽  
Bar Resolution

We construct comparison morphisms between two well-known projective resolutions of a monomial algebra $A$: the bar resolution $\operatorname{\mathbb{Bar}} A$ and Bardzell's resolution $\operatorname{\mathbb{Ap}} A$; the first one is used to define the cup product and the Lie bracket on the Hochschild cohomology $\operatorname{HH} ^*(A)$ and the second one has been shown to be an efficient tool for computation of these cohomology groups. The constructed comparison morphisms allow us to show that the cup product restricted to even degrees of the Hochschild cohomology has a very simple description. Moreover, for $A= \mathbb{k} Q/I$ a monomial algebra such that $\dim_ \mathbb{k} e_i A e_j = 1$ whenever there exists an arrow $\alpha: i \to j \in Q_1$, we describe the Lie action of the Lie algebra $\operatorname{HH}^1(A)$ on $\operatorname{HH}^{\ast} (A)$.

scholarly journals Complete slices and homological properties of tilted algebras

1994 ◽  
Vol 36 (3) ◽  
pp. 347-354 ◽  
Author(s):  
Ibrahim Assem ◽  
Flávio Ulhoa Coelho
Keyword(s):  
Representation Theory ◽  
Global Dimension ◽  
Indecomposable Modules ◽  
Tilted Algebra ◽  
Finite Dimensional ◽  
Tilted Algebras ◽  
Homological Dimensions ◽  
The Subject ◽  
Left And Right ◽  
Almost All

It is reasonable to expect that the representation theory of an algebra (finite dimensional over a field, basic and connected) can be used to study its homological properties. In particular, much is known about the structure of the Auslander-Reiten quiver of an algebra, which records most of the information we have on its module category. We ask whether one can predict the homological dimensions of a module from its position in the Auslander-Reiten quiver. We are particularly interested in the case where the algebra is a tilted algebra. This class of algebras of global dimension two, introduced by Happel and Ringel in [7], has since then been the subject of many investigations, and its representation theory is well understood by now (see, for instance, [1], [7], [8], [9], [11], [13]).In this case, the most striking feature of the Auslander-Reiten quiver is the existence of complete slices, which reproduce the quiver of the hereditary algebra from which the tilted algebra arises. It follows from well-known results that any indecomposable successor (or predecessor) of a complete slice has injective (or projective, respectively) dimension at most one, from which one deduces that a tilted algebra is representation-finite if and only if both the projective and the injective dimensions of almost all (that is, all but at most finitely many non-isomorphic) indecomposable modules equal two (see (3.1) and (3.2)). On the other hand, the authors have shown in [2, (3.4)] that a representation-infinite algebra is concealed if and only if both the projective and the injective dimensions of almost all indecomposable modules equal one (see also [14]). This leads us to consider, for tilted algebras which are not concealed, the case when the projective (or injective) dimension of almost all indecomposable successors (or predecessors, respectively) of a complete slice equal two. In order to answer this question, we define the notions of left and right type of a tilted algebra, then those of reduced left and right types (see (2.2) and (3.4) for the definitions).

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