Ibrahim Assem
The objective of this paper is to give a concrete interpretation of the dimension of the first Hochschild cohomology space of a cyclically oriented or tame cluster tilted algebra in terms of a numerical invariant arising from the potential.
Let C be a finite dimensional algebra of global dimension at most two. A partial relation extension is any trivial extension of C by a direct summand of its relation C-C-bimodule. When C is a tilted algebra, this construction provides an…
This short note is devoted to motivate and clarify the notion of sequential walk introduced by the authors in a previous work. We also give some applications of this concept.
We show how to compute the low Hochschild cohomology groups of a partial relation extension algebra.
We prove that the representation dimension of finite dimensional selfinjective algebras over a field is invariant under socle equivalence and derive some consequences.
We give necessary and sufficient conditions for torsion pairs in a hereditary category to be in bijection with $t$-structures in the bounded derived category of that hereditary category. We prove that the existence of a split $t$-structure…
We prove that the representation dimension of a selfinjective algebra of wild tilted type is equal to three, and give an explicit construction of an Auslander generator of its module category. We also show that if a connected selfinjective…
We characterize the indecomposable transjective modules over an arbitrary cluster-tilted algebra that do not lie on a local slice, and we provide a sharp upper bound for the number of (isoclasses of) these modules.
In this paper, we prove that relation-extensions of quasi-tilted algebras are 2-Calabi-Yau tilted. With the objective of describing the module category of a cluster-tilted algebra of euclidean type, we define the notion of reflection so…
Let $A$ be a special biserial algebra over an algebraically closed field. We show that the first Hohchshild cohomology group of $A$ with coefficients in the bimodule $A$ vanishes if and only if $A$ is representation finite and simply…
Let $B$ be the split extension of a finite dimensional algebra $C$ by a $C$-$C$-bimodule $E$. We define a morphism of associative graded algebras $\varphi^*:\HH^*(B)\rightarrow \HH^*(C)$ from the Hochschild cohomology of $B$ to that of $C$,…
We prove that the representation dimension of a selfinjective algebra of euclidean type is equal to three, and give an explicit construction of the Auslander generator of its module category.
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, then we show that if C is constrained, or…
In this article, we introduce the notion of cluster automorphism of a given cluster algebra as a $\ZZ$-automorphism of the cluster algebra that sends a cluster to another and commutes with mutations. We study the group of cluster…
In this paper, we introduce a notion of unistructural cluster algebras, for which the set of cluster variables uniquely determines the clusters. We prove that cluster algebras of Dynkin type and cluster algebras of rank 2 are unistructural,…
Given a cluster-tilted algebra B we study its first Hochschild cohomology group HH1(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 E= Ext2(DC,C), then we prove that HH1(B)…
We prove that indecomposable transjective modules over cluster-tilted algebras are uniquely determined by their dimension vectors. Similarly, we prove that for cluster-concealed algebras, rigid modules lifting to rigid objects in the…
We introduce a category of cluster algebras with fixed initial seeds. This category has countable coproducts, which can be constructed combinatorially, but no products. We characterise isomorphisms and monomorphisms in this category and…
Given a finite dimensional algebra A of finite global dimension, we consider the trivial extension of A by the A-A-bimodule $\oplus_{i\ge 2} \Ext^2_A(DA,A)$, which we call the higher relation bimodule. We first give a recipe allowing to…
We give a criterion allowing to verify whether or not two tilted algebras have the same relation-extension (thus correspond to the same cluster-tilted algebra). This criterion is in terms of a combinatorial configuration in the…