Piotr Malicki

Finite cycles of indecomposable modules

This work was completed with the support of the research grant DEC-2011/02/A/ST1/00216 of the Polish National Science Center and the CIMAT Guanajuato, Mexico.

ON AUSLANDER-REITEN COMPONENTS OF ALGEBRAS WITHOUT EXTERNAL SHORT PATHS

The research supported by the Research Grant N N201 269135 of the Polish Ministry of Science and Higher Education

On the number of terms in the middle of almost split sequences over cycle-finite artin algebras

We prove that the number of terms in the middle of an almost split sequence in the module category of a cycle-finite artin algebra is bounded by 5.

Cycle-Finite Module Categories

We describe the structure of module categories of finite dimensional algebras over an algebraically closed field for which the cycles of nonzero nonisomorphisms between indecomposable finite dimensional modules are finite (do not belong to the infinite Jacobson radical of the module category). Moreover, geometric and homological properties of these module categories are exhibited.

Algebras with separating Auslander–Reiten components

Cluster categories have been introduced by Buan, Marsh, Reineke, Reiten and Todorov in order to categorify Fomin-Zelevinsky cluster algebras. This survey motivates and outlines the construction of a generalization of cluster categories, and explains different applications of these new categories in representation theory.This survey article is intended as an introduction to the recent categorical classification theorems of the three authors, restricting to the special case of the category of modules for a finite group.Let k be a field. A finite dimensional k-algebra is said to be minimal representation-infinite provided it is representation-infinite and all its proper factor algebras are representation-finite. Our aim is to classify the special biserial algebras which are minimal representation-infinite. The second part describes the corresponding module categories.These are expanded notes from three survey lectures given at the 14th International Conference on Representations of Algebras (ICRA XIV) held in Tokyo in August 2010. We first study identities between products of quantum dilogarithm series associated with Dynkin quivers following Reineke. We then examine similar identities for quivers with potential and link them to Fomin-Zelevinskys theory of cluster algebras. Here we mainly follow ideas due to Bridgeland, Fock-Goncharov, Kontsevich-Soibelman and Nagao.These notes reflect the contents of three lectures given at the workshop of the 14th International Conference on Representations of Algebras (ICRA XIV), held in August 2010 in Tokyo. We first provide an introduction to quantum loop algebras and their finite-dimensional representations. We explain in particular Nakajimas geometric description of the irreducible q-characters in terms of graded quiver varieties. We then present a recent attempt to understand the tensor structure of the category of finite- dimensional representations by means of cluster algebras.

Cycle-Finite Algebras with Finitely Many τ-Rigid Indecomposable Modules

We prove that every cycle-finite artin algebra with finitely many isomorphism classes of τ-rigid indecomposable modules is of finite representation type.

MODULES NOT BEING THE MIDDLE OF SHORT CHAINS

We give a complete description of finitely generated modules over artin algebras which are not the middle of a short chain of modules, using injective and tilting modules over hereditary artin algebras.

Auslander-Reiten components determined by their composition factors

The research supported by the Research Grant N N201 269135 of the Polish Ministry of Science and Higher Education.

On The Number of Simple and Projective Modules in the Quasi-Tubes of Self-Injective Algebras

We establish a bound on the number of simple and projective modules in the quasi-tubes of the Auslander–Reiten quivers of finite dimensional self-injective algebras over a field.

Infinite Components of An Auslander-Reiten Quiver with Finite DTr-Orbits

Abstract Let A be a basic connected finite dimensional algebra over an algebraically closed field. We show that if Γ is an infinite connected component of the Auslander-Reiten quiver ΓA of A in which each ΓA-orbit contains only finitely many vertices, then the number of indecomposable direct summands of the middle term of any mesh, whose starting vertex belongs to the infinite stable part of Γ, is less than or equal to 3. Moreover, if the nonstable vertices belong to τA-orbits of exceptional projectives in Γ, then Γ can be obtained from a stable tube by a finite number of multiple coray-ray insertions of type α*γ and multiple coray-ray insertions of type α*γ.