Peter Dräxler

Exact categories and vector space categories

In a series of papers additive subbifunctors F of the bifunctor ExtΛ( , ) are studied in order to establish a relative homology theory for an artin algebra Λ. On the other hand, one may consider the elements of F (X, Y ) as short exact sequences. We observe that these exact sequences make mod Λ into an exact category if and only if F is closed in the sense of Butler and Horrocks. Concerning the axioms for an exact category we refer to Gabriel and Roiter’s book. In fact, for our general results we work with subbifunctors of the extension functor for arbitrary exact categories. In order to study projective and injective objects for exact categories it turns out to be convenient to consider categories with almost split exact pairs, because many earlier results can easily be adapted to this situation. Exact categories arise in representation theory for example if one studies categories of representations of bimodules. Representations of bimodules gained their importance in studying questions about representation types. They appear as domains of certain reduction functors defined on categories of modules. These reduction functors are often closely related to the functor ExtΛ( , ) and in general do not preserve at all the usual exact structure of mod Λ. By showing the closedness of suitable subbifunctors of ExtΛ( , ) we can equip mod Λ with an exact structure such that some reduction functors actually become ‘exact’. This allows us to derive information about the projective and injective objects in the respective categories of representations of bimodules appearing as domains, and even show that almost split sequences for them. Examples of such domains appearing in practice are the subspace categories of a vector space category with bonds. We provide an example showing that existence of almost split sequences for them is not a general fact but may even fail if the vector space category is finite.

Towards the classification of sincere weakly positive unit forms

Abstract Unit forms occur in the representation theory of algebras, partially ordered sets and related areas. They appear as Tits forms and Euler forms. Their weak positivity resp. non-negativity frequently characterizes finite and tame representation types. Their positive roots correspond to indecomposable representations. Let e(n) be the vector in Z n with co-ordinates of which are all 1. We show how all weakly positive unit forms can be reconstructed from those unit forms such that e(n) is a root and such that there is no radical vector μ ≠ 0, where ϵ(n) ± μ is a positive root as well. We call unit forms with these properties ‘good thin forms’. Our main result is the complete classification of all good thin unit forms, which rests on the purely combinatorial construction of all these forms in few variables by using a computer.

Biextensions by Indecomposable Modules of Derived Regular Length 2

We establish sufficient conditions for a biextension algebra of a piecewise hereditary algebra of type Ān or Dn by indecomposable modules of derived regular length 2 to be of tame representation type.

Domestic Algebras with Many Nonperiodic Auslander–Reiten Components

Abstract We exhibit a wide class of domestic finite dimensional algebras over an algebraically closed field whose Auslander–Reiten quivers admit infinitely many connected components of type ℤ𝔻∞ (respectively, of types ℤ𝔻∞ and ℤ ). The algebras are suitable iterated one-point extensions of hereditary algebras of Euclidean type 𝔸˜ n .

On indecomposable modules over directed algebras

Generalizing a result of Bongartz we show that any nonsimple indecomposable module over a finite-dimensional k-algebra A is an extension of an indecomposable and a simple module provided k is a field with more than two elements and A is representation directed. Our proof is based on fibre sums over simple modules and some known classification results on socle projective modules over peak algebras. In case the global dimension of A is at most 2 our methods also yield a description of the dimension vectors of the indecomposable ,4-modules by the roots of the associated quadratic form.

Representation-Directed Diamonds

A module over a finite-dimensional algebra is called a ‘diamond’ if it has a simple top and a simple socle.Using covering theory, the classification of all diamonds for algebras of finite representation type over algebraically closed fields can be reduced to representation-directed algebras. We prove a criterion referring to the positive roots of the corresponding Tits quadratic form, which makes it easy to check whether a representation-directed algebra has a faithful diamond. Using an implementation of this criterion in the CREP program system on representation theory, we are able to classify all exceptional representation-directed algebras having a faithful diamond. We obtain a list of 157 algebras up to isomorphism and duality. The 52 maximal members of this list are presented at the end of this paper.

Tree algebras with non-negative tits form

One of the seminal results of modern representation theory of algebras is Gabriels theorem saying that a quiver algebra A = kQ accepts only finitely many isoclasses of indecomposable modules if and only if the underlying quiver of Q is a Dynkin diagram. The proof may be done by introducing the Tits form q ~ of A and showing that the conditions above are equivalent to qa being a positive quadratic form. In fact, the isoclasses of indecomposable A-modules are in one-to-one correspondence with the positive roots of q ~ . Some years later it was shown that a quiver algebra A = kQ is of tame type if and only if the corresponding Tits form is non-negative. See [8]. We recall that a finite dimensional algebra over an algebraically closed field k is said to be tame if for each dimension, almost all the indecomposable modules occur in a finite number of one-parameter families. We are interested in the classification of tame algebras and their representations. Let A be a basic triangular algebra with a presentation A = kQ/I, where Q is a finite connected quiver without oriented cycles and I is an admissible ideal of the path algebra kQ, see [8]. Let Qo = (1,. . . , n) be the set of vertices of Q. The Tits form q ~ : Zn+ Z is defined by

Cleaving functors and controlled wild algebras

Abstract We show that the cleaving functors introduced in [Bautista et al., Invent. Math. 81 (1985) 217] as a tool for proving infinite representation type of finite-dimensional algebras can also be used to establish controlled wildness. The main application is that an algebra is controlled wild if there is an indecomposable projective module with a Loewy factor having a homogeneous direct summand which is of length at least 3. As a second application we derive Hans covering criterion.

Normal Forms for Representations of Representation-finite Algebras

Using the CREP system we show that matrix representations of representation-finite algebras can be transformed into normal forms consisting of (0, 1)-matrices.

Derived dynkin extensions

The finite dimensional tame hereditary algebras are associated with the extended Dynkin diagrams. An indecomposable module over such an algebra is either preprojective or preinjective or lies in a family of tubes whose tubular type is the corresponding Dynkin diagram. The study of one-point extensions by simple regular modules in such tubes was initiated in [Ri]. We generalise this approach by starting out with algebras which are derived equivalent to a tame hereditary algebra and considering one-point extensions by modules which are simple regular in tubes in the derived category. If the obtained tubular type is again a Dynkin diagram these algebras are called derived Dynkin extensions. Our main theorem says that a representation infinite algebra is derived equivalent to a tame hereditary algebra iff it is an iterated derived Dynkin extension of a tame concealed algebra. As application we get a new proof of a theorem in [AS] about domestic tubular branch enlargements which uses the derived category inste...