Δ-Filtrations and Projective Resolutions for the Auslander–Dlab–Ringel Algebra (original) (raw)

Auslander Algebras as Quasi-Hereditary Algebras

Journal of the London Mathematical Society, 1989

The notion of a quasi-hereditary algebra has been introduced by E. Cline, B. Parshall and L. Scott [7,2,5] in order to describe the so-called highest weight categories arising in the representation theory of Lie algebras and algebraic groups. Quasi-hereditary algebras are defined by the existence of a suitable chain of ideals, and the finite dimensional hereditary algebras are typical examples. In [3], also finite dimensional algebras of global dimension 2 are shown to be quasi-hereditary. Thus, the Auslander algebras are quasi-hereditary. Recall that the Auslander algebras A can be constructed in the following way. Let R be a representation-finite finite dimensional algebra; then A is the endomorphism algebra End(Af fl), where M is a finite dimensional /{-module such that every indecomposable /{-module is isomorphic to a direct summand of M. We are going to introduce the notion of a splitting filtration on the class of all indecomposable /{-modules and show that in this way we obtain a heredity chain of ideals of A (see the definition below). Usually, there exist many splitting filtrations for a given R. Examples of splitting filtrations can be obtained from the Rojter measure, used by A. V. Rojter in his proof of the first Brauer-Thrall conjecture [6], or from the preprojective and preinjective partitions, introduced by M. Auslander and S. Smalo in [1]. Instead of dealing with finite dimensional algebras, we shall consider, more generally, semiprimary rings. Recall that an associative ring A with 1 is called semiprimary provided that its Jacobson radical TV is nilpotent and A/N is semisimple artinian. We say that an ideal J of A is a heredity ideal of A if P = J, JNJ = 0 and /, considered as a right ideal, is a projective ^-module. Following [2], a semiprimary ring A is said to be quasi-hereditary provided that there exists a chain 0 = 7 0 <= 4 c ... c ^ <= ^ <= ... c: 4 = ,4 of ideals of A such that, for each 1 < t < m, the ideal J t /J t _ x is a heredity ideal of A/J t _ v Such a sequence of ideals is called a heredity chain. Some elementary facts related to these concepts can be found in [3]. Splitting filtrations Let R be a semiprimary ring of finite representation type. (Of course, this implies that R is both left artinian and right artinian.) We denote by indR a complete set of representatives of the isomorphism classes of indecomposable modules. If M c ind R, then add M denotes the full additive subcategory of mod R generated by M. Its objects are direct sums of copies of modules in M. A direct sum of a copies of a module X will be denoted by aX.

The left part and the Auslander-Reiten components of an artin algebra

2005

The left part L A of the module category of an artin algebra A consists of all indecomposables whose predecessors have projective dimension at most one. In this paper, we study the Auslander-Reiten components of A (and of its left support A λ) which intersect L A and also the class E of the indecomposable Ext-injectives in the addditive subcategory addL A generated by L A .

Gorenstein-Projective Modules over Upper Triangular Matrix Artin Algebras

Journal of Mathematics

Gorenstein-projective module is an important research topic in relative homological algebra, representation theory of algebras, triangulated categories, and algebraic geometry (especially in singularity theory). For a given algebra A , how to construct all the Gorenstein-projective A -modules is a fundamental problem in Gorenstein homological algebra. In this paper, we describe all complete projective resolutions over an upper triangular Artin algebra Λ = A M B A 0 B . We also give a necessary and sufficient condition for all finitely generated Gorenstein-projective modules over Λ = A M B A 0 B .

A note on preprojective partitions over hereditary Artin algebras

Proceedings of the American Mathematical Society, 1982

If A is an artin algebra there is a partition of ind A, the category of indecomposable finitely generated A-modules, ind A = U^g/V, called the prepro-* jective partition. We show that Pl can be easily constructed for hereditary artin algebras, if P,_ ¡

Filtrations and completions of certain positive level modules of affine algebras

Advances in Mathematics, 2005

This paper was motivated by an effort to understand the representation theoretic meaning of the results of on realizations of (pseudo-)crystal bases of certain quantum loop modules in the framework of Littelmann's path model. These papers showed in particular, that one could write the tensor product of a crystal basis of a highest weight integrable module with a (pseudo-)crystal basis of such a quantum loop module as a union of highest weight crystals. The obvious and natural intepretation would be that the decomposition of the crystals gave rise to a direct sum decomposition of the tensor product of the corresponding modules for the quantum affine algebra. It is however, not very difficult to see that such a tensor product never contains a copy of a highest weight module. In addition, the corresponding classical situation which was studied in [7] and more recently in [1, 20] did not exclude the possibility that such tensor products might in fact be irreducible. In this paper we are able to show that the tensor product of an integrable highest weight representation with the quantum loop module associated to the natural representation admits a filtration such that the successive quotients are highest weight integrable modules with multiplicity and highest weight given by the path model.

Projective resolutions of associative algebras and ambiguities

Journal of Algebra, 2015

The aim of this article is to give a method to construct bimodule resolutions of associative algebras, generalizing Bardzell's well-known resolution of monomial algebras. We stress that this method leads to concrete computations, providing thus a useful tool for computing invariants associated to the algebras. We illustrate how to use it giving several examples in the last section of the article. In particular we give necessary and sufficient conditions for noetherian down-up algebras to be 3-Calabi-Yau.

The Ringel dual of the Auslander–Dlab–Ringel algebra

Journal of Algebra

The ADR algebra R A of a finite-dimensional algebra A is a quasihereditary algebra. In this paper we study the Ringel dual R(R A) of R A. We prove that R(R A) can be identified with (R A op) op , under certain 'minimal' regularity conditions for A. We also give necessary and sufficient conditions for the ADR algebra to be Ringel selfdual.