With the help of Galois coverings, we describe the tame tensor products $A{\otimes }_{K}B$ of basic, connected, nonsimple, finite-dimensional algebras A and B over an algebraically closed field K. In particular, the description of all tame group algebras AG of finite groups G over finite-dimensional algebras A is completed.

A criterion for tame prinjective type for a class of posets with zero-relations is given in terms of the associated prinjective Tits quadratic form and a list of hypercritical posets. A consequence of this result is that if ${\Lambda }^{•}$ is a three-partite subamalgam of a tiled order then it is of tame lattice type if and only if the reduced Tits quadratic form $q_{{\Lambda }^{•}}$ associated with ${\Lambda }^{•}$ in [26] is weakly non-negative. The result generalizes a criterion for tameness of such orders given by Simson [28] and gives an...

We give a description of faces, of all codimensions, for the cones spanned by the set of weights associated to the rings of semi-invariants of quivers. For a triple flag quiver and its faces of codimension 1 this description reduces to the result of Knutson-Tao-Woodward on the facets of the Klyachko cone. We give new applications to Littlewood-Richardson coefficients, including a product formula for LR-coefficients corresponding to triples of partitions lying on a wall of the Klyachko cone. We systematically...

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 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 $({\Gamma }_Q,G)$.

This paper studies the Hochschild cohomology of finite-dimensional monomial algebras. If Λ = K/I with I an admissible monomial ideal, then we give sufficient conditions for the existence of an embedding of $K[x₁,...,x_r]/⟨x_a{x}_{b}fora\ne b⟩$ into the Hochschild cohomology ring HH*(Λ). We also introduce stacked algebras, a new class of monomial algebras which includes Koszul and D-Koszul monomial algebras. If Λ is a stacked algebra, we prove that $HH*\left(\Lambda \right)/\cong K[x₁,...,x_r]/⟨x_a{x}_{b}fora\ne b⟩$, where is the ideal in HH*(Λ) generated by the homogeneous nilpotent elements. In...

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.

Let Γ be a finite-dimensional hereditary basic algebra. We consider the radical rad Γ as a Γ-bimodule. It is known that there exists a quasi-hereditary algebra 𝓐 such that the category of matrices over rad Γ is equivalent to the category of Δ-filtered 𝓐-modules ℱ(𝓐,Δ). In this note we determine the quasi-hereditary algebra 𝓐 and prove certain properties of its module category.