Andrzej Skowroński

On Auslander-Reiten components of blocks and self-injective biserial algebras

We investigate the existence of Auslander-Reiten components of Euclidean type for special biserial self-injective algebras and for blocks of group algebras. In particular we obtain a complete description of stable Auslander-Reiten quivers for the tame self-injective algebras considered here

Minimal representation-infinite coil algebras

For a basic and connected finite dimensional algebra A over an algebraically closed field, we study when the cycles in the category mod A (of finite dimensional modules) are well-behaved. We call A cycle-finite if, for any cycle in mod A, no morphism on the cycle lies in the infinite power of the radical. We show that, in this case, A is tame. We also introduce a natural generalisation of a tube, called a coil, and define A to be a coil algebra if any cycle in mod A lies in a standard coil. We prove that the minimal representation-infinite coil algebras coincide with the tame concealed algebras.

Deformed preprojective algebras of generalized Dynkin type

We introduce the class of deformed preprojective algebras of generalized Dynkin graphs An (n > 1), D n (n > 4), EG, E 7 , Eg and L n (n ≥ 1) and prove that it coincides with the class of all basic connected finite-dimensional self-injective algebras for which the inverse Nakayama shift v -1 S of every non-projective simple module S is isomorphic to its third syzygy Ω 3 S.

Group algebras of polynomial growth

Let A be a finite dimensional, basic and connected algebra (associative, with 1) over an algebraically closed field, G a finite group and AG the group algebra. The main result gives necessary and sufficient conditions for AG to be of polynomial growth, that is, there is a natural number m such that the indecomposable finite dimensional AG-modules occur, in each dimension d≧2, in a finite number of discrete and at most d one-parameter families.

The algebras of semi-invariants of quivers

We show that the algebras of semi-invariants of a finite connected quiverQ are complete intersections if and only ifQ is of Dynkin or Euclidean type. Moreover, we give a uniform description of the algebras of semi-invariants of Euclidean quivers.

Cycles in Module Categories

Let A be an artin algebra over a commutative artin ring R and mod A be the category of finitely generated right A-modules. A cycle in mod A is a sequence of non-zero non-isomorphisms M 0 → M 1 → ... → M n = M 0 between indecomposable modules from mod A. The main aim of this survey article is to show that study of cycles in mod A leads to interesting information on indecomposable A-modules, the Auslander-Reiten quiver of A, and the ring structure of A. We present recent advances in some areas of the representation theory of artin algebras which should be of interest to a wider audience. In the paper, we also pose a number of open problems and indicate some new research directions.

Classification of discrete derived categories

The main aim of the paper is to classify the discrete derived categories of bounded complexes of modules over finite dimensional algebras.

Characterizations of algebras with small homological dimensions

Abstract We introduce the class of double tilted algebras, containing the class of tilted algebras and prove various characterizations. In particular, we show that the class of double tilted algebras is the class of all artin algebras whose AR-quiver admits a faithful double section with a natural property. Moreover, we prove that the class of double tilted algebras coincides with the class of all artin algebras of global dimension three, for which every indecomposable finitely generated module has projective or injective dimension at most one. We also describe the structure of the category of finitely generated modules as well as the AR-quiver of double tilted algebras.

Cycle-finite algebras

Abstract Let A be a finite-dimensional K -algebra over an algebraically closed field K and mod A be the category of finitely generated right A -modules. Following [1], A is said to be cycle-finite if, for every cycle M 0 → m 1 → … → M n = M 0 of non-zero non-isomorphisms between indecomposable modules in mod A , the morphisms on this cycle do not belong to the infinite power of the Jacobson radical of mod A . In this article we describe the supports of stable tubes of the Auslander-Reiten quivers of cycle-finite algebras. As a consequence we get that every cycle-finite algebra is of polynomial growth. Moreover, we prove some characterizations of domestic cycle-finite algebras.

Short chains and short cycles of modules

We show that for a large class of artin algebras including the algebras of finite representation type, an indecomposable module M is not the middle of a short chain if and only if there is no short cycle M -N -M of nonzero nonisomorphisms between indecomposable modules. We apply this to get sufficient conditions for modules to be determined by their composition factors. We also show that if for an algebra of finite representation type there is a sincere indecomposable A-module that is not the middle of a short chain, then A is a tilted algebra.