Radical cube zero selfinjective algebras of finite complexity

One of our main results is a classification of all the possible quivers of selfinjective radical cube zero finite-dimensional algebras over an algebraically closed field having finite complexity. In the paper (Erdmann and Solberg, 2011) [5] we classified all weakly symmetric algebras with support varieties via Hochschild cohomology satisfying Dade’s Lemma. For a finite-dimensional algebra to have such a theory of support varieties implies that the algebra has finite complexity. Hence this paper is a partial extension of [5].     

Artin-Schelter regular algebras and categories

Motivated by constructions in the representation theory of finite dimensional algebras we generalize the notion of Artin-Schelter regular algebras of dimension n to algebras and categories to include Auslander algebras and a graded analogue for infinite representation type. A generalized Artin-Schelter regular algebra or a category of dimension n is shown to have common properties with the classical Artin-Schelter regular algebras. In particular, when they admit a duality, then they satisfy Serre duality formulas and the -category of nice sets of simple objects of maximal projective dimension n is a finite length Frobenius category.     

Weighted locally gentle quivers and Cartan matrices

We introduce and study the class of weighted locally gentle quivers. This naturally extends the class of gentle quivers and gentle algebras, which have been intensively studied in the representation theory of finite-dimensional algebras, to a wider class of potentially infinite-dimensional algebras. Weights on the arrows of these quivers lead to gradings on the corresponding algebras. For natural grading by path lengths, any locally gentle algebra is Koszul. The class of locally gentle algebras consists of the gentle algebras together with their Koszul duals.Our main result is a general combinatorial formula for the determinant of the weighted Cartan matrix of a weighted locally gentle quiver. We show that this weighted Cartan determinant is a rational function which is completely determined by the combinatorics of the quiver-more precisely by the number and the weight of certain oriented cycles.     

Graded and Koszul Categories

Koszul algebras have arisen in many contexts; algebraic geometry, combinatorics, Lie algebras, non-commutative geometry and topology. The aim of this paper and several sequel papers is to show that for any finite dimensional algebra there is always a naturally associated Koszul theory. To obtain this, the notions of Koszul algebras, linear modules and Koszul duality are extended to additive (graded) categories over a field. The main focus of this paper is to provide these generalizations and the necessary preliminaries.     

One characteristical property of semilattices

The operations of node deletion and insertion in a finite dimensional quiver algebra were introduced in Martínez-Villa (1980) as an abstraction of the operations used in earlier works (Auslander and Reiten, 1973; Bongartz and Riedtmann, 1979; Platzeck, 1978), such constructions are the easiest way to produce stably equivalent algebras.

In general, it is not easy to decide whether or not a given quadratic algebra is Koszul, then it is of interest to construct new Koszul algebras from given ones. The aim of the article is to prove that node deletion and insertion generalizes to graded quiver algebras producing, as in the finite dimensional case, stably equivalent algebras and, in this Situation, either both or neither of the two algebras are Koszul.     

Twisted support varieties

We define and study twisted support varieties for modules over an Artin algebra, where the twist is induced by an automorphism of the algebra. Under a certain finite generation hypothesis we show that the twisted variety of a module satisfies Dade’s Lemma and is one dimensional precisely when the module is periodic with respect to the twisting automorphism. As a special case we obtain results on DTr-periodic modules over Frobenius algebras.     

Tilting equivalences: From hereditary algebras to symmetric groups

We study Koszul duality for finite dimensional hereditary algebras, and various generalisations to trivial extension algebras, to Schur algebras, to doubles of Schur bialgebras, and to deformations of doubles of Schur bialgebras. We describe applications to the modular representation theory of symmetric groups.     

Support variety theory is invariant under singular equivalences of Morita type

Using a new equivalent definition of support varieties in the sense of Snashall and Solberg [23], we show that both the (Fg) condition and support varieties are preserved under singular equivalences of Morita type. In particular, support variety theory is invariant under stable equivalences of Morita type.     

Rational homotopy theory and differential graded category

We propose a generalization of Sullivan’s de Rham homotopy theory to non-simply connected spaces. The formulation is such that the real homotopy type of a manifold should be the closed tensor dg-category of flat bundles on it much the same as the real homotopy type of a simply connected manifold is the de Rham algebra in original Sullivan’s theory. We prove the existence of a model category structure on the category of small closed tensor dg-categories and as a most simple case, confirm an equivalence between the homotopy category of spaces whose fundamental groups are finite and whose higher homotopy groups are finite dimensional rational vector spaces and the homotopy category of small closed tensor dg-categories satisfying certain conditions.     

Noetherianity and Ext

Let R=R₀⊕R₁⊕R₂⊕? be a graded algebra over a field K such that R₀ is a finite product of copies of K and each R_i is finite dimensional over K. Set J=R₁⊕R₂⊕? and . We study the properties of the categories of graded R-modules and S-modules that relate to the noetherianity of R. We pay particular attention to the case when R is a Koszul algebra and S is the Koszul dual to R.     

Complex group algebras of finite groups: Brauer's Problem 1

Brauer's Problem 1 asks the following: What are the possible complex group algebras of finite groups? It seems that with the present knowledge of representation theory it is not possible to settle this question. The goal of this paper is to present a partial solution to this problem. We conjecture that if the complex group algebra of a finite group does not have more than a fixed number m of isomorphic summands, then its dimension is bounded in terms of m. We prove that this is true for every finite group if it is true for the symmetric groups. The problem for symmetric groups reduces to an explicitly stated question in number theory or combinatorics.     

BGP-reflection functors and cluster combinatorics

We define Bernstein-Gelfand-Ponomarev reflection functors in the cluster categories of hereditary algebras. They are triangle equivalences which provide a natural quiver realization of the “truncated simple reflections” on the set of almost positive roots Φ_≥−1 associated with a finite dimensional semi-simple Lie algebra. Combining this with the tilting theory in cluster categories developed in [A. Buan, R. Marsh, M. Reineke, I. Reiten, G. Todorov, Tilting theory and cluster combinatorics, Adv. Math. (in press). math.RT/0402054], we give a unified interpretation via quiver representations for the generalized associahedra associated with the root systems of all Dynkin types (simply laced or non-simply laced). This confirms the Conjecture 9.1 in [A. Buan, R. Marsh, M. Reineke, I. Reiten, G. Todorov, Tilting theory and cluster combinatorics, Adv. Math. (in press). math.RT/0402054] for all Dynkin types.     

Periodic resolutions and self-injective algebras of finite type

We say that an algebra A is periodic if it has a periodic projective resolution as an (A,A)-bimodule. We show that any self-injective algebra of finite representation type is periodic. To prove this, we first apply the theory of smash products to show that for a finite Galois covering B→A, B is periodic if and only if A is. In addition, when A has finite representation type, we build upon results of Buchweitz to show that periodicity passes between A and its stable Auslander algebra. Finally, we use Asashiba’s classification of the derived equivalence classes of self-injective algebras of finite type to compute bounds for the periods of these algebras, and give an application to stable Calabi-Yau dimensions.     

An elementary approach to Wedderburn's structure theory

Wedderburn's theorem on the structure of finite dimensional (semi)simple algebras is proved by using minimal prerequisites.     

How to construct a ‘concrete’ superdecomposable pure-injective module over a string algebra

We construct an element in a direct product of finite dimensional modules over a string algebra such that the pure-injective envelope of this element is a superdecomposable module.     

A note on stable equivalences of Morita type

We investigate when an exact functor --Γ which induces a stable equivalence is part of a stable equivalence of Morita type. If Λ and Γ are finite dimensional algebras over a field k whose semisimple quotients are separable, we give a necessary and sufficient condition for this to be the case. This generalizes a result of Rickard’s for self-injective algebras. As a corollary, we see that the two functors given by tensoring with the bimodules in a stable equivalence of Morita type are right and left adjoints of one another, provided that these bimodules are indecomposable. This fact has many interesting consequences for stable equivalences of Morita type. In particular, we show that a stable equivalence of Morita type induces another stable equivalence of Morita type between certain self-injective algebras associated to the original algebras. We further show that when there exists a stable equivalence of Morita type between Λ and Γ, it is possible to replace Λ by a Morita equivalent k-algebra Δ such that Γ is a subring of Δ and the induction and restriction functors induce inverse stable equivalences.     

Mixed algebras and quivers related to cell complexes

In this paper we study finite dimensional algebras arising from categories of perverse sheaves on finite regular cell complexes (cellular perverse algebras). We prove that such algebras are quasi-hereditary and have finite global dimension. We discuss some restrictions, under which cellular perverse algebras are Koszul. We also study the relationship between Koszul duality functors in the derived categories of categories of graded and non-graded modules over an algebra and its quadratic dual.     

Ghosts in modular representation theory

A ghost over a finite p-group G is a map between modular representations of G which is invisible in Tate cohomology. Motivated by the failure of the generating hypothesis—the statement that ghosts between finite-dimensional G-representations factor through a projective—we define the ghost number of kG to be the smallest integer l such that the composite of any l ghosts between finite-dimensional G-representations factors through a projective. In this paper we study ghosts and the ghost numbers of p-groups. We begin by showing that a weaker version of the generating hypothesis, where the target of the ghost is fixed to be the trivial representation k, holds for all p-groups. We then compute the ghost numbers of all cyclic p-groups and all abelian 2-groups with C₂ as a summand. We obtain bounds on the ghost numbers for abelian p-groups and for all 2-groups which have a cyclic subgroup of index 2. Using these bounds we determine the finite abelian groups which have ghost number at most 2. Our methods involve techniques from group theory, representation theory, triangulated category theory, and constructions motivated from homotopy theory.     

Topological groups and covering dimension

We consider a natural way of extending the Lebesgue covering dimension to various classes of infinite dimensional topological groups. The dimension function that we introduce extends Lebesgue covering dimension, has the hereditary property, and has a product theory that is more similar to the product theory for the finite dimensional case.