Stratified Algebras Research Articles

Tame and wild theorem for the category of modules filtered by standard modules

We introduce the notion of interlaced weak ditalgebras and apply reduction procedures to their module categories to prove a tame-wild dichotomy for the category F(Δ) of Δ-filtered modules for an arbitrary finite homological system (P,≤,{Δi}i∈P). This includes the case of standardly stratified algebras. Moreover, in the tame case, we show that given a fixed dimension d, for every d-dimensional indecomposable module M∈F(Δ), with the only possible exception of those lying in a finite number of isomorphism classes, the module M coincides with its Auslander-Reiten translate in F(Δ). Our proofs rely on the equivalence of F(Δ) with the module category of some special type of ditalgebra.

Semi-Infinite Highest Weight Categories

We develop axiomatics of highest weight categories and quasi-hereditary algebras in order to incorporate two semi-infinite situations which are in Ringel duality with each other; the underlying posets are either upper finite or lower finite. We also consider various more general sorts of stratified categories. In the upper finite cases, we give an alternative characterization of these categories in terms of based quasi-hereditary algebras and based stratified algebras, which are certain locally unital algebras possessing triangular bases.

Mixed standardization and Ringel duality

Dlab–Ringel's standardization method gives a realization of a standardly stratified algebra. In this paper, we construct mixed stratified algebras, which are a generalization of standardly stratified algebras, following Dlab–Ringel's standardization method. Moreover, we study a Ringel duality of mixed stratified algebras from the viewpoint of stratifying systems.

Essential orders on stratified algebras with duality and S-subcategories in O

We prove uniqueness of the essential order for stratified algebras having simple preserving duality, generalizing a recent result of Coulembier for quasi-hereditary algebras. We apply this to classify, up to equivalence, regular integral blocks of S-subcategories in the BGG category O. We also describe various homological invariants of these blocks.

On properly stratified Gorenstein algebras

We show that a properly stratified algebra is Gorenstein if and only if the characteristic tilting module coincides with the characteristic cotilting module. We further show that properly stratified Gorenstein algebras A enjoy strong homological properties such as all Gorenstein projective modules being properly stratified and all endomorphism rings End A ( Δ ( i ) ) being Frobenius algebras. We apply our results to the study of properly stratified algebras that are minimal Auslander-Gorenstein algebras in the sense of Iyama-Solberg and calculate under suitable conditions their Ringel duals. This applies in particular to all centraliser algebras of nilpotent matrices.

A GENERALIZATION OF THE THEORY OF STANDARDLY STRATIFIED ALGEBRAS I: STANDARDLY STRATIFIED RINGOIDS

AbstractWe extend the classical notion of standardly stratified k-algebra (stated for finite dimensional k-algebras) to the more general class of rings, possibly without 1, with enough idempotents. We show that many of the fundamental results, which are known for classical standardly stratified algebras, can be generalized to this context. Furthermore, new classes of rings appear as: ideally standardly stratified and ideally quasi-hereditary. In the classical theory, it is known that quasi-hereditary and ideally quasi-hereditary algebras are equivalent notions, but in our general setting, this is no longer true. To develop the theory, we use the well-known connection between rings with enough idempotents and skeletally small categories (ringoids or rings with several objects).

Permutation modules for cellularly stratified algebras

Permutation modules play an important role in the representation theory of the symmetric group. Hartmann and Paget defined permutation modules for Brauer algebras. We generalise their construction to a wider class of algebras, namely cellularly stratified algebras, satisfying certain conditions. We give a decomposition into indecomposable summands, the Young modules, and show that permutation modules and Young modules admit cell filtrations (with well-defined filtration multiplicities). Partition algebras are shown to satisfy the given conditions, provided the characteristic of the underlying field is large enough. Thus we obtain a definition of permutation modules for partition algebras as an application.

Inductions and restrictions on towers of cellularly stratified diagram algebras

Hartmann et al. defined the concept of cellularly stratified algebras that combine the features of both cellular algebras and stratified algebras. Many important diagram algebras in mathematics and physics, such as some Brauer, partition and BMW algebras, are cellularly stratified algebras, and each of these forms a tower of algebras. This article gives the concept of towers of cellularly stratified algebras in an axiomatic manner, and studies it in terms of induction and restriction functors. In particular, for certain towers of cellularly stratified algebras, we provide a criterion for semi-simplicity by using the cohomology groups of cell modules.

STRATIFYING SYSTEMS FOR EXACT CATEGORIES

AbstractIn this paper, we develop the theory of stratifying systems in the context of exact categories as a generalisation of the notion of stratifying systems in module categories, which have been studied by different authors. We prove that attached to a stratifying system in an exact category $(\mathcal{A},\mathcal{E})$ there is an standardly stratified algebra B such that the category $\mathscr{F}$F(Θ), of F-filtered objects in the exact category $(\mathcal{A},\mathcal{E})$ is equivalent to the category $\mathscr{F}$(Δ) of Δ-good modules associated to B. The theory we develop in exact categories, give us a way to produce standardly stratified algebras from module categories by just changing the exact structure on it. In this way, we can construct exact categories whose bounded derived category is equivalent to the bounded derived category of an standardly stratified algebra. Finally, applying the relative homological algebra developed by Auslander–Solberg, we can construct examples of stratifying systems that are not a stratifying system in the classical sense, so our approach really produces new stratifying systems.

Wide subcategories of finitely generated Λ-modules

We explore some properties of wide subcategories of the category [Formula: see text] of finitely generated left [Formula: see text]-modules, for some artin algebra [Formula: see text] In particular we look at wide finitely generated subcategories and give a connection with the class of standard modules and standardly stratified algebras. Furthermore, for a wide class [Formula: see text] in [Formula: see text] we give necessary and sufficient conditions to see that [Formula: see text] for some projective [Formula: see text]-module [Formula: see text] and finally, a connection with ring epimorphisms is given.

On minimal Auslander–Gorenstein algebras and standardly stratified algebras

We give new properties of algebras with finite self-injective dimension coinciding with the dominant dimension d≥2, which are called minimal Auslander–Gorenstein algebras in the recent work of Iyama and Solberg, see [22]. In particular, when those algebras are standardly stratified, we give criteria when the category of (properly) (co)standardly filtered modules has a nice homological description using tools from the theory of dominant dimensions and Gorenstein homological algebra. We give some examples of standardly stratified algebras having dominant dimension equal to the self-injective dimension, including examples having an arbitrary natural number as the self-injective dimension and blocks of finite representation type of Schur algebras.

Local and global methods in representations of Hecke algebras

This paper aims at developing a "local--global" approach for various types of finite dimensional algebras, especially those related to Hecke algebras. The eventual intention is to apply the methods and applications developed here to the cross-characteristic representation theory of finite groups of Lie type. The authors first review the notions of quasi-hereditary and stratified algebras over a Noetherian commutative ring. They prove that many global properties of these algebras hold if and only if they hold locally at every prime ideal. When the commutative ring is sufficiently good, it is often sufficient to check just the prime ideals of height at most one. These methods are applied to construct certain generalized q-Schur algebras, proving they are often quasi-hereditary (the "good" prime case) but always stratified. Finally, these results are used to prove a triangular decomposition matrix theorem for the modular representations of Hecke algebras at good primes. In the bad prime case, the generalized q-Schur algebras are at least stratified, and a block triangular analogue of the good prime case is proved, where the blocks correspond to Kazhdan-Lusztig cells.

English

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.

Stratifying endomorphism algebras using exact categories

This paper constructs enlargements of Hecke algebras over Z[t,t−1] to certain standardly stratified algebras. The latter are obtained as endomorphism algebras of modules with dual left cell module filtrations in the sense of Kazhdan–Lusztig. A novel feature of the proofs is the use of suitably chosen exact categories to avoid difficult Ext1-vanishing conditions.

Homological Systems in Triangulated Categories

We introduce the notion of homological systems Θ for triangulated categories. Homological systems generalize, on one hand, the notion of stratifying systems in module categories, and on the other hand, the notion of exceptional sequences in triangulated categories. We prove that, attached to the homological system Θ, there are two standardly stratified algebras A and B, which are derived equivalent. Furthermore, it is proved that the category $\mathfrak {F}({\Theta }),$ of the Θ-filtered objects in a triangulated category $\mathcal {T},$ admits in a very natural way a structure of an exact category, and then there are exact equivalences between the exact category $\mathfrak {F}({\Theta })$ and the exact categories of the Δ-good modules associated to the standardly stratified algebras A and B. Some of the obtained results can be seen also under the light of the cotorsion pairs in the sense of Iyama-Nakaoka-Yoshino (see 6.6 and 6.7 ). We recall that cotorsion pairs are studied extensively in relation with cluster tilting categories, t-structures and co-t-structures.

A generalized Koszul theory and its relation to the classical theory

Let A=⨁i⩾0Ai be a graded locally finite k-algebra where A0 is a finite dimensional algebra whose finitistic dimension is 0. In this paper we develop a generalized Koszul theory preserving many classical results, and show an explicit correspondence between this generalized theory and the classical theory. Applications in representations of certain categories and extension algebras of standard modules of standardly stratified algebras are described.

Split-By-Nilpotent Extensions Algebras and Stratifying Systems

Let Γ and Λ be artin algebras such that Γ is a split-by-nilpotent extension of Λ by a two sided ideal I of Γ. Consider the change of rings functors G: =ΓΓΛ ⊗Λ − and F: =ΛΛΓ ⊗Γ −. In this article, by assuming that I Λ is projective, we find the necessary and sufficient conditions under which a stratifying system (Θ, ≤) in modΛ can be lifted to a stratifying system (GΘ, ≤) in mod(Γ). Furthermore, by using the functors F and G, we study the relationship between their filtered categories of modules; and some connections with their corresponding standardly stratified algebras are stated (see Theorem 5.12, Theorem 5.15 and Theorem 5.18). Finally, a sufficient condition is given for stratifying systems in mod(Γ) in such a way that they can be restricted, through the functor F, to stratifying systems in mod(Λ).

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.

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.

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.