Erzsébet Lukács

Finitistic dimension of standardly stratified algebras

We prove that the projectively and the injectively defined finitistic dimensions of a standardly stratified algebra are always finite by giving the optimal bound for these numbers in terms of the number of simple modules.

Standardly Stratified Extension Algebras

Abstract The paper generalizes some of our previous results on quasi-hereditary Koszul algebras to graded standardly stratified Koszul algebras. The main result states that the class of standardly stratified algebras for which the left standard modules as well as the right proper standard modules possess a linear projective resolution – the so called linearly stratified algebras – is closed under forming their Yoneda extension algebras. This is proved using the technique of Hilbert and Poincare series of the corresponding modules. #Communicated by D. Happel.

Minimal relation algebras

This paper is concerned with the covers of the atoms in the lattice of varieties of relation algebras. Aminimal relation algebra is one that is simple and generates such a subvariety. The main result we prove is that there are exactly three finite minimal relation algebras that aretotally symmetric (i.e., satisfy the identitiesx=x andx ≤ x; x). We also give an example of an infinite minimal totally symmetric relation algebra, and some results about other subvarieties.

Constructions of Stratified Algebras

In this article, a construction to build recursively all basic finite dimensional standardly stratified algebras is given. In comparison to the construction described by Dlab and Ringel for the quasi-hereditary case ([15]) some new features appear here.

Homological characterization of lean algebras

Certain classes of lean quasi-hereditary algebras play a central role in the representation theory of semisimple complex Lie algebras and algebraic groups. The concept of a lean semiprimary ring, introduced recently in [1] is given here a homological characterization in terms of the surjectivity of certain induced maps between Ext1-groups. A stronger condition requiring the surjectivity of the induced maps between Extk-groups for allk≥1, which appears in the recent work of Cline, Parshall and Scott on Kazhdan-Lusztig theory, is shown to hold for a large class of lean quasi-hereditary algebras.

STRATIFYING PAIRS OF SUBCATEGORIES FOR CPS-STRATIFIED ALGEBRAS

Two special types of module subcategories are defined over stratified algebras of Cline, Parshall and Scott. We show that for every stratified algebra there exists a (not necessarily unique) cotorsion pair of subcategories which describe to a large extent the stratification structure of the algebra. These subcategories generalize the notion of modules with standard and costandard filtration for standardly stratified and quasi-hereditary algebras.

Well-filtered algebras

We define a class of (lean) quasi-hereditary K-algebras A for which the standard filtration of the right regular representation may be described by a suitable directed quotient algebra A+. For this class, projective resolutions of simple left modules over A− will correspond to the so-called BGG resolutions over A, defined earlier by Bernstein, Gelfand and Gelfand. In the case when K is algebraically closed and A+ is a subalgebra of A, A+ coincides with the concept of a Borel subalgebra of Konig. We show that many algebras obtained by previously defined canonical constructions belong to this class and have additional structural properties.

Stratified modules over an extension algebra

Let A be a standard Koszul standardly stratified algebra and X an A-module. The paper investigates conditions which imply that the module Ext* A(X) over the Yoneda extension algebra A* is filtered by standard modules. In particular, we prove that the Yoneda extension algebra of A is also standardly stratified. This is a generalization of similar results on quasi-hereditary and on graded standardly stratified algebras.

Standard Koszul standardly stratified algebras

ABSTRACT In this paper, we prove that every standard Koszul (not necessarily graded) standardly stratified algebra is also Koszul. This generalizes a similar result of [3] on quasi-hereditary algebras.

Construction of CPs-Stratified Algebras

The results of [7] and [2] gave a recursive construction for all quasi-hereditary and standardly stratified algebras starting with local algebras and suitable bimodules. Using the notion of stratifying pairs of subcategories, introduced in [3], we generalize these earlier results to construct recursively all CPS-stratified algebras.