We describe all finite-dimensional algebras A over an algebraically closed field for which the algebra $T_2\left(A\right)$ of 2×2 upper triangular matrices over A is of tame representation type. Moreover, the algebras A for which $T_2\left(A\right)$ is of polynomial growth (respectively, domestic, of finite representation type) are also characterized.

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 ${\Sigma }_A$.

We study when the composite of n irreducible morphisms between modules in a regular component of the Auslander-Reiten quiver is non-zero and lies in the n+1-th power of the radical ℜ of the module category. We prove that in this case such a composite belongs to ${\Re }^{\infty }$. We apply these results to characterize those string algebras having n irreducible morphisms between band modules such that their composite is a non-zero morphism in ${\Re }^{n+1}$.

Let $Q$ be a finite union of Dynkin quivers, $G\subseteq \mathrm{Aut}\left(𝕜Q\right)$ a finite abelian group, $\widehat{Q}$ the generalized McKay quiver of $(Q,G)$ and ${\Gamma }_Q$ the Auslander-Reiten quiver of $𝕜Q$. Then $G$ acts functorially on the quiver ${\Gamma }_Q$. We show that the Auslander-Reiten quiver of $𝕜\widehat{Q}$ coincides with the generalized McKay quiver of $({\Gamma }_Q,G)$.

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.

A ring Λ satisfies the Generalized Auslander-Reiten Condition ( ) if for each Λ-module M with $Ex{t}^i(M,M\oplus \Lambda )=0$ for all i > n the projective dimension of M is at most n. We prove that this condition is satisfied by all n-symmetric algebras of quasitilted type.

A class of finite-dimensional algebras whose Auslander-Reiten quivers have starting but not generalized standard components is investigated. For these components the slices whose slice modules are tilting are considered. Moreover, the endomorphism algebras of tilting slice modules are characterized.

Using geometrical methods, Huisgen-Zimmermann showed that if M is a module with simple top, then M has no proper degeneration $M{<}_{deg}N$ such that ${}^{t}M{/}^{t+1}M{\simeq }^{t}N{/}^{t+1}N$ for all t. Given a module M with square-free top and a projective cover P, she showed that $di{m}_{k}Hom(M,M)=di{m}_{k}Hom(P,M)$ if and only if M has no proper degeneration $M{<}_{deg}N$ where M/M ≃ N/N. We prove here these results in a more general form, for hom-order instead of degeneration-order, and we prove them algebraically. The results of Huisgen-Zimmermann follow as consequences from our results....

Let A be a finite-dimensional algebra over a field k. We discuss the existence of trisections (mod₊ A,mod₀ A,mod₋ A) of the category of finitely generated modules mod A satisfying exactness, standardness, separation and adjustment conditions. Many important classes of algebras admit trisections. We describe a construction of algebras admitting a trisection of their module categories and, in special cases, we describe the structure of the components of the Auslander-Reiten quiver lying in mod₀ A.