We prove that a completely separating incidence algebra of a partially ordered set is of tame representation type if and only if the associated Tits integral quadratic form is weakly non-negative.

We prove that the component quiver ${\Sigma }_A$ of a connected self-injective artin algebra A of infinite representation type is fully cyclic, that is, every finite set of components of the Auslander-Reiten quiver ${\Gamma }_A$ of A lies on a common oriented cycle in ${\Sigma }_A$.

Given a module M over a domestic canonical algebra Λ and a classifying set X for the indecomposable Λ-modules, the problem of determining the vector $m\left(M\right)={\left(m_x\right)}_{x\in X}\in {\mathbb{N}}^X$ such that $M\cong {⨁}_{x\in X}{X}_x^{m_x}$ is studied. A precise formula for $di{m}_{k}Ho{m}_{\Lambda }(M,X)$, for any postprojective indecomposable module X, is computed in Theorem 2.3, and interrelations between various structures on the set of all postprojective roots are described in Theorem 2.4. It is proved in Theorem 2.2 that a general method of finding vectors m(M) presented by the authors in Colloq....

Given a module M over an algebra Λ and a complete set of pairwise nonisomorphic indecomposable Λ-modules, the problem of determining the vector $m\left(M\right)={\left(m_X\right)}_{X\in }\in {\mathbb{N}}^{}$ such that $M\cong {⨁}_{X\in }{X}^{m_X}$ is studied. A general method of finding the vectors m(M) is presented (Corollary 2.1, Theorem 2.2 and Corollary 2.3). It is discussed and applied in practice for two classes of algebras: string algebras of finite representation type and hereditary algebras of type ${̃}_{p,q}$. In the second case detailed algorithms are given (Algorithms 4.5 and 5.5).

The Dynkin algebras are the hereditary artin algebras of finite representation type. The paper determines the number of complete exceptional sequences for any Dynkin algebra. Since the complete exceptional sequences for a Dynkin algebra of Dynkin type Δ correspond bijectively to the maximal chains in the lattice of non-crossing partitions of type Δ, the calculations presented here may also be considered as a categorification of the corresponding result for non-crossing partitions.

We describe the representation-infinite blocks B of the group algebras KG of finite groups G over algebraically closed fields K for which all simple modules are periodic with respect to the action of the syzygy operators. In particular, we prove that all such blocks B are periodic algebras of period 4. This confirms the periodicity conjecture for blocks of group algebras.

Auslander’s representation dimension measures how far a finite dimensional algebra is away from being of finite representation type. In [1], M. Auslander proved that a finite dimensional algebra A is of finite representation type if and only if the representation dimension of A is at most 2. Recently, R. Rouquier proved that there are finite dimensional algebras of an arbitrarily large finite representation dimension. One of the exciting open problems is to show that all finite dimensional algebras...

Let S be a commutative local ring of characteristic p, which is not a field, S* the multiplicative group of S, W a subgroup of S*, G a finite p-group, and $S^{\lambda }G$ a twisted group ring of the group G and of the ring S with a 2-cocycle λ ∈ Z²(G,S*). Denote by $In{d}_m\left(S^{\lambda }G\right)$ the set of isomorphism classes of indecomposable $S^{\lambda }G$-modules of S-rank m. We exhibit rings $S^{\lambda }G$ for which there exists a function $f_{\lambda }:\mathbb{N}\to \mathbb{N}$ such that $f_{\lambda }\left(n\right)\ge n$ and $In{d}_{f_{\lambda }\left(n\right)}\left(S^{\lambda }G\right)$ is an infinite set for every natural n > 1. In special cases $f_{\lambda }\left(\mathbb{N}\right)$ contains every natural number m >...

We prove that for any representation-finite algebra A (in fact, finite locally bounded k-category), the universal covering F: Ã → A is either a Galois covering or an almost Galois covering of integral type, and F admits a degeneration to the standard Galois covering F̅: Ã→ Ã/G, where $G=\Pi \left({\Gamma }_A\right)$ is the fundamental group of ${\Gamma }_A$. It is shown that the class of almost Galois coverings F: R → R’ of integral type, containing the series of examples from our earlier paper [Bol. Soc. Mat. Mexicana 17 (2011)], behaves...

We classify the irreducible components of varieties of modules over tubular algebras. Our results are stated in terms of root combinatorics. They can be applied to understand the varieties of modules over the preprojective algebras of Dynkin type 𝔸₅ and 𝔻₄.