Based on the work of D. Happel, I. Reiten and S. Smalø on quasitilted artin algebras, the first two authors recently introduced the notion of quasitilted rings. Various authors have presented examples of quasitilted artin algebras that are not tilted. Here we present a class of right quasitilted rings that not right tilted, and we show that they satisfy a condition that would force a quasitilted artin algebra to be tilted.

In continuation of our article in Colloq. Math. 116.1, we give a complete description of the symmetric algebras of strictly canonical type by quivers and relations, using Brauer quivers.

We construct arbitrarily complicated indecomposable finite-dimensional modules with periodic syzygies over symmetric algebras.

Over an artinian hereditary ring R, we discuss how the existence of almost split sequences starting at the indecomposable non-injective preprojective right R-modules is related to the existence of almost split sequences ending at the indecomposable non-projective preinjective left R-modules. This answers a question raised by Simson in [27] in connection with pure semisimple rings.

We investigate the category $\text{mod}\Lambda$ of finite length modules over the ring $\Lambda =A{\otimes }_k\Sigma$, where $\Sigma$ is a V-ring, i.e. a ring for which every simple module is injective, $k$ a subfield of its centre and $A$ an elementary $k$-algebra. Each simple module $E_j$ gives rise to a quasiprogenerator $P_j=A\otimes E_j$. By a result of K. Fuller, $P_j$ induces a category equivalence from which we deduce that $\text{mod}\Lambda \simeq {\coprod }_{j}badhbox{P}_j$. As a consequence we can (1) construct for each elementary $k$-algebra $A$ over a finite field $k$ a nonartinian noetherian ring $\Lambda$ such that $\text{mod}A\simeq \text{mod}\Lambda$, (2) find twisted...

We provide a characterization of artin algebras without chains of nonzero homomorphisms between indecomposable finitely generated modules starting with an injective module and ending with a projective module.

We discuss the existence of tilting modules which are direct limits of finitely generated tilting modules over tilted algebras.

The aim of this work is to characterize the algebras which are standardly stratified with respect to any order of the simple modules. We show that such algebras are exactly the algebras with all idempotent ideals projective. We also deduce as a corollary a characterization of hereditary algebras, originally due to Dlab and Ringel.

Let A be an Artin algebra and let $0\to X\to {\oplus }_{i=1}^r{Y}_i\to Z\to 0$ be an almost split sequence of A-modules with the $Y_i$ indecomposable. Suppose that X has a projective predecessor and Z has an injective successor in the Auslander-Reiten quiver ${\Gamma }_A$ of A. Then r ≤ 4, and r = 4 implies that one of the $Y_i$ is projective-injective. Moreover, if $X\to {\oplus }_{j=1}^t{Y}_j$ is a source map with the $Y_j$ indecomposable and X on an oriented cycle in ${\Gamma }_A$, then t ≤ 4 and at most three of the $Y_j$ are not projective. The dual statement for a sink map holds. Finally, if an arrow...

We construct bar-invariant $\mathbb{Z}\left[q^{\pm 1/2}\right]$-bases of the quantum cluster algebra of the valued quiver $A_2^{\left(2\right)}$, one of which coincides with the quantum analogue of the basis of the corresponding cluster algebra discussed in P. Sherman, A. Zelevinsky: Positivity and canonical bases in rank 2 cluster algebras of finite and affine types, Moscow Math. J., 4, 2004, 947–974.

We describe the stable module categories of the self-injective finite-dimensional algebras of finite representation type over an algebraically closed field which are Calabi-Yau (in the sense of Kontsevich).

In a highly influential paper, Bidigare, Hanlon and Rockmore showed that a number of popular Markov chains are random walks on the faces of a hyperplane arrangement. Their analysis of these Markov chains took advantage of the monoid structure on the set of faces. This theory was later extended by Brown to a larger class of monoids called left regular bands. In both cases, the representation theory of these monoids played a prominent role. In particular, it was used to compute the spectrum of the...

We describe the structure of artin algebras for which all cycles of indecomposable finitely generated modules are finite and all Auslander-Reiten components are semiregular.

We describe the structure of artin algebras for which all cycles of indecomposable modules are finite and almost all indecomposable modules have projective or injective dimension at most one.

We establish when the partial orders ${\le }_{ext}$ and ${\le }_{deg}$ coincide for all modules of the same dimension from the additive category of a generalized standard almost cyclic coherent component of the Auslander-Reiten quiver of a finite-dimensional algebra.