We determine the Hochschild cohomology of all finite-dimensional generalized multicoil algebras over an algebraically closed field, which are the algebras for which the Auslander-Reiten quiver admits a separating family of almost cyclic coherent components. In particular, the analytically rigid generalized multicoil algebras are described.

We consider the socle deformations arising from formal deformations of a class of Koszul self-injective special biserial algebras which occur in the study of the Drinfeld double of the generalized Taft algebras. We show, for these deformations, that the Hochschild cohomology ring modulo nilpotence is a finitely generated commutative algebra of Krull dimension 2.

The incidence coalgebras $K^{\square }I$ of interval finite posets I and their comodules are studied by means of the reduced Euler integral quadratic form $q^{•}:{\mathbb{Z}}^{\left(I\right)}\to \mathbb{Z}$, where K is an algebraically closed field. It is shown that for any such coalgebra the tameness of the category $K^{\square }I-comod$ of finite-dimensional left $K^{\square }I$-modules is equivalent to the tameness of the category $K^{\square }I-Como{d}_{fc}$ of finitely copresented left $K^{\square }I$-modules. Hence, the tame-wild dichotomy for the coalgebras $K^{\square }I$ is deduced. Moreover, we prove that for an interval finite ̃ *ₘ-free...

Almost completely decomposable groups with a critical typeset of type $(1,3)$ and a $p$-primary regulator quotient are studied. It is shown that there are, depending on the exponent of the regulator quotient $p^k$, either no indecomposables if $k\le 2$; only six near isomorphism types of indecomposables if $k=3$; and indecomposables of arbitrary large rank if $k\ge 4$.

We describe the structure of all indecomposable modules in standard coils of the Auslander-Reiten quivers of finite-dimensional algebras over an algebraically closed field. We prove that the supports of such modules are obtained from algebras with sincere standard stable tubes by adding braids of two linear quivers. As an application we obtain a complete classification of non-directing indecomposable modules over all strongly simply connected algebras of polynomial growth.

We discuss the problem of classification of indecomposable representations for extended Dynkin quivers of type 𝔼̃₈, with a fixed orientation. We describe a method for an explicit determination of all indecomposable preprojective and preinjective representations for those quivers over an arbitrary field and for all indecomposable representations in case the field is algebraically closed. This method uses tilting theory and results about indecomposable modules for a canonical algebra of type (5,3,2)...

In this paper we study the precise relation between two representations of a given split finite basic dimensional algebra A as a factor of the free path algebra over its quiver (A). After defining the notion of strongly acyclic quiver, we apply the results obtained to develop a method of calculating the group Aut(A)/Inn(A) in the case when (A) is strongly acyclic.

Let Λ be a finite-dimensional, basic and connected algebra over an algebraically closed field, and mod Λ be the category of finitely generated right Λ-modules. We say that Λ has acceptable projectives if the indecomposable projective Λ-modules lie either in a preprojective component without injective modules or in a standard coil, and the standard coils containing projectives are ordered. We prove that for such an algebra Λ the following conditions are equivalent: (a) Λ is tame, (b) the Tits form...

We prove that a stably hereditary bound quiver algebra A = KQ/I is iterated tilted if and only if (Q,I) satisfies the clock condition, and that in this case it is of type~Q. Furthermore, A is tilted if and only if (Q,I) does not contain any double-zero.

In this article we study the interplay between algebro-geometric notions related to $\pi$-points and structural features of the stable Auslander-Reiten quiver of a finite group scheme. We show that $\pi$-points give rise to a number of new invariants of the AR-quiver on one hand, and exploit combinatorial properties of AR-components to obtain information on $\pi$-points on the other. Special attention is given to components containing Carlson modules, constantly supported modules, and endo-trivial modules.

We prove that the Krull-Gabriel dimension of the category of modules over any 1-domestic non-degenerate string algebra is 3.

We study a connection between left-right projective bimodules and stable equivalences of Morita type for finite-dimensional associative algebras over a field. Some properties of the category of all finite-dimensional left-right projective bimodules for self-injective algebras are also given.

Bimodules over triangular Nakayama algebras that give stable equivalences of Morita type are studied here. As a consequence one obtains that every stable equivalence of Morita type between triangular Nakayama algebras is a Morita equivalence.

Let K be a field and Γ a finite quiver without oriented cycles. Let Λ := K(Γ,ρ) be the quotient algebra of the path algebra KΓ by the ideal generated by ρ, and let 𝒟(Λ) be the dual extension of Λ. We prove that each Lie derivation of 𝒟(Λ) is of the standard form.

Let Λ be a directed finite-dimensional algebra over a field k, and let B be an upper triangular bimodule over Λ. Then we show that the category of B-matrices mat B admits a projective generator P whose endomorphism algebra End P is quasi-hereditary. If A denotes the opposite algebra of End P, then the functor Hom(P,-) induces an equivalence between mat B and the category ℱ(Δ) of Δ-filtered A-modules. Moreover, any quasi-hereditary algebra whose category of Δ-filtered modules is equivalent to mat...