G-semisimple algebras

Almost split sequences in dimension two

Cohen–Macaulay modules, (co)torsion pairs and virtually Gorenstein algebras

Methods and results from the representation theory of finite di- mensional algebras have led to many interactions with other areas of mathe- matics. Such areas include the theory of Lie algebras and quantum groups, commutative algebra, algebraic geometry and topology, and in particular the new theory of cluster algebras. The aim of this workshop was to further de- velop such interactions and to stimulate progress in the representation theory of algebras.

Almost split sequences and Auslander-Reiten quivers (species) have turned out to be an extremely useful tool in the representation theory of artinian algebras and integral representations of complete orders. Whereas the theory is well developed in the artinian situations, it is far behind for integral representations. The main reason is that the powerful tool of universal coverings and covering techniques — as developed by Riedtmann [14], Bongartz — Gabriel [4] — has not yet found its integral analogue. The main difficulty is that it is unclear to me, when a representation quiver — in the sense of C.Riedtmann [14] — is the Auslander-Reiten quiver of an order of finite lattice type; in particular, one obstacle is the multiplication with the parameter of the ground ring. A little progress has been made in this direction by Wiedemann. Apart from my general philosophy, that most of the results in the representation theory of artinian algebras have more or less obvious analoga in integral representation theory, there is a more solid connection between integral representations and some categories of torsionfree modules over artinian algebras as observed by C.M.Ringel and the author [15], such that in some instances integral representation theory might also influence the representation theory of artinian algebras — e.g. that some torsionfree categories of artinian modules have almost split sequences was — before its general treatment by M.Auslander and S.O.Smalo [1] — in secial cases — influenced by the results in integral representation theory — noted by the author [17] and independently by R.Bautista and R.Martinez [3].

Let Λ be an Artin algebra, let modΛ be the category of finitely generated Λ-modules, and let A⊂modΛ be a contravariantly finite and extension closed subcategory. For an indecomposable and not Ext-projective module C∈A, we compute the almost split sequence 0→A→B→C→0 in A from the almost split sequence 0→D TrC→E→C→0 in modΛ. Since the computation is particularly simple if the minimal right A-approximation of D TrC is indecomposable for all indecomposable and not Ext-projective C∈A, we manufacture subcategories A with the desired property using orthogonal subcategories. The method of orthogonal subcategories is applied to compute almost split sequences for relatively projective and prinjective modules.

For an Artin Algebra Λ of finite representation type, the category Λ-mod, considered as a ring with several objects, has Krull dimension zero. Contrary, for a wild hereditary Artin Algebra this dimension does not exist. In this paper we show that the Krull dimension of Λ-mod for an Artin Algebra Λ of tame representation type is two. The corresponding Krull-Gabriel filtration by Serre subcategories of the category F of finitely presented contravariant functors on Λ-mod leads to a hierarchy of exact sequences in Λ-mod. Influenced by the functorial approach to almost split sequences by M. Auslander and I. Reiten, we investigate the exact sequences whose corresponding functors become simple in one of the successive quotient categories of the Krull-Gabriel filtration of F.

We study algebraic and homological properties of two classes of infinite-dimensional Hopf algebras over an algebraically closed field k k of characteristic 0 0 . The first class consists of those Hopf k k -algebras that are connected graded as algebras, and the second class are those Hopf k k -algebras that are connected as coalgebras. For many but not all of the results presented here, the Hopf algebras are assumed to have finite Gel’fand–Kirillov dimension. It is shown that if the Hopf algebra H H is a connected graded Hopf algebra of finite Gel’fand–Kirillov dimension n n , then H H is a noetherian domain which is Cohen–Macaulay, Artin–Schelter regular, and Auslander regular of global dimension n n . It has S 2 = I d H S^2 = \mathrm {Id}_H , and it is Calabi–Yau. Detailed information is also provided about the Hilbert series of H H . Our results leave open the possibility that the first class of algebras is (properly) contained in the second. For this second class, the Hopf k k -algebras of finite Gel’fand–Kirillov dimension n n with connected coalgebra, the underlying coalgebra is shown to be Artin–Schelter regular of global dimension n n . Both these classes of Hopf algebras share many features in common with enveloping algebras of finite-dimensional Lie algebras. For example, an algebra in either of these classes satisfies a polynomial identity only if it is a commutative polynomial algebra. Nevertheless, we construct, as one of our main results, an example of a Hopf k k -algebra H H of Gel’fand–Kirillov dimension 5, which is connected graded as an algebra and connected as a coalgebra, but is not isomorphic as an algebra to U ( g ) U(\mathfrak {g}) for any Lie algebra g \mathfrak {g} .

We have already pointed out in Sect.1.5 that Auslander–Reiten quivers provide a threefold information about the representation theory of the quiver, namely the indecomposable representations, the irreducible morphisms, and the almost split sequences—these in turn should be thought of the building blocks of arbitrary representations, morphisms, and short exact sequences, respectively.KeywordsAuslander-Reiten QuiverIndecomposable RepresentationsCoxeter ElementFinite Representation TypePath SectionThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

An Artin algebra Λ is said to be of finite Cohen–Macaulay type if, up to isomorphism, there are only finitely many indecomposable modules in G(Λ), the full subcategory of modΛ consisting of all Gorenstein projective (right) Λ-modules. In this paper, we study 1-Gorenstein algebras of finite Cohen–Macaulay type through mod(G(Λ)), the category of finitely presented G(Λ)-modules. Some applications will be provided. In particular, a necessary and sufficient condition is given for T3(Λ), the 3 by 3 lower triangular matrices over Λ, to be of finite Cohen–Macaulay type. Finally, the structure of almost split sequences will be described explicitly in a special subcategory of mod(G(Λ)), denoted by ϑ−1(G(Λ)). If Λ is self-injective, ϑ−1(G(Λ))=mod(G(Λ)).

Let R be a ring, see below for other notation. The functor categories (mod-R, Ab) and ((R-mod)op, Ab) have received considerable attention since the 1960s. The first of these has achieved prominence in the model theory of modules and most particularly in the investigation of the representation theory of Artinian algebras. Both [11, Chapter 12] and [8] contain accounts of the use (mod-R, Ab) may be put to in the model theoretic setting, and Auslander's review, [1], details the application of (mod-R, Ab) to the study of Artinian algebras. The category ((R-mod)op, Ab) has been less fully exploited. Much work, however, has been devoted to the study of the transpose functor between R-mod and mod-R. Warfield's paper, [13], describes this for semiperfect rings, and this duality is an essential component in the construction of almost split sequences over Artinian algebras, see [4]. In comparison, the general case has been neglected. This paper seeks to remedy this situation, giving a concrete description of the resulting equivalence between (mod-R, Ab) and ((R-mod)op, Ab) for an arbitrary ring R.

Quadratic Lie algebras with 2-plectic structures

Matlis duality and the particular case of Macaulay correspondence provide a dictionary between the Artin algebras and their inverse systems. Inspired in a result of Emsalem we translate the problem of classification of Artin algebras to a problem of linear system of equations on the inverse systems.

On subcategories closed under predecessors and the representation dimension

In this paper, kreb algebras are introduced. It is shown that that the class of kreb algebras is a wider class than the class of BCI algebras. Properties of kreb algebras are presented. A new binary relation different from the one in classical BCI algebras is defined in kreb algebras, and it is shown that with this relation, kreb algebras are partially ordered sets. It is established that this relation is not a partial ordering in classical BCI algebras. The partial ordering is then used to investigate other properties of kreb algebras. New kinds of ideals and homomorphisms are also introduced and studied in kreb algebras independent of these concepts in classical BCI algebras.

Consider a set I which, in the text which follows, will play the role of a universal set. The different subsets of I form a set on which may be defined three operations: union, intersection and complementation. Endowed with these, the set of subsets of I then constitutes an ‘algebra’, usually known as an “algebra of sets” or an “algebra of classes”. From the definitions of set, subset and those of the three operations above the diverse properties of this algebra can be demonstrated. The more important of these properties will be stated later. They will be justified with the aid of Venn diagrams. However, by way of an exercise the reader can carry out rigorous abstract demonstrations based on the definitions in Chapter 1. It will be shown by an example (relationship E9 below) which type of demonstration can then be made.