# Test problems for the theory of finite dimensional algebras (10w5069)

Arriving in Banff, Alberta Sunday, September 12 and departing Friday September 17, 2010

## Organizers

José Antonio de la Peña (Centro de Investigación en Matemáticas)

Vlastimil Dlab (Carleton University)

Helmut Lenzing (University of Paderborn)

Claus Michael Ringel (Universitaet Bielefeld)

## Objectives

To check the status of a sufficiently mature mathematical theory, late Irving Kaplansky has coined the concept of a "test problem" (in the context of infinite abelian groups).

The proposed Workshop intends to apply such tests to a number of directions by focussing on the following questions:

1. The representation theory of wild Kronecker algebras. The $r$-Kronecker algebra $K_r$ is the path algebra of the quiver

$ 1 defr{rightarrow} renewcommandarraystretch{0}begin{array}{c}

r r vdots r

end{array} 2$ ($r$ arrows). For each $r$ the algebra $K_r$ is hereditary, thus belongs to that class of algebras whose representation theory is best understood. For $r=1$ the representation type of $K_r$ is finite, for $r=2$ it is tame (and derived equivalent to the classification of coherent sheaves on the projective line), while for $rgeq 3$ the classification problem is wild, and generally thought to be hopeless.

We propose to attack this problem by means of emph{covering theory}. In the hands of Gabriel and his school, covering theory was very successfully applied to finite type and later with similar success extended to tame type by Dowbor-Skowronski. By now it is a standard tool to investigate tame algebras and their representations, but curiously a serious application of covering methods for wild type does not exist. For instance covering theory plays a key role in the recently completed classification of tame self-injective algebras (satisfying some natural extra conditions) by Skowronski and collaborators. Coming back to the proposed topic, we expect that attacking the representation theory of wild Kronecker algebras by covering theory will yield substantial new insight, provide a significant stimulus to further development, in particular introduce a fresh point of view in the study of representations for algebras of wild type in general. Because of the fundamental importance of the representation theory of the Kronecker quiver and its appearance in many situations in mathematics, we expect new lines of interactions to neighboring subjects as a natural byproduct of this investigation.

2. The unknown nature of Kerner's exotic space. For a wild hereditary $k$-algebra the indecomposable modules fall into three classes: the indecomposable preprojective (resp. preinjective) modules, each forming a single, hence countable, Auslander-Reiten component, and an infinite family of regular components, all of the same combinatorial shape $ZZAA_infty$. A fundamental, and very surprising, result of O. Kerner states that there are natural bijections (in general not unique) between the sets of regular components for any two (connected) wild hereditary algebras, thus giving rise to a unique universal parametrizing space $Omega$. The mathematical structure of this space is completely unknown, even conjecturally. $Omega$ should be thought of to be a space of quite exotic nature.

It is curious that the space $Omega$ occurs in other mathematical contexts as well, since $Omega$ also parametrizes

the set of regular components of a wild canonical algebra, or the set of Auslander-Reiten components of vector bundles on a weighted projective line of Euler characteristic $<0$, or --- again surprisingly --- the set of Auslander-Reiten components for the triangulated category $mathcal{D}^b_{sg}(R)$ of singularities for (graded) surface singularities of Brieskorn or Fuchsian type.

Our aim to put this as a topic for the intended workshop is twofold: first we want to complete the picture, showing that further algebras and categories yield the 'same' parametrizing space and second we want to collect (partial or full) evidence about its nature.

3. Sequences of algebras. Many classes of finite dimensional algebras have been extensively studied. For instance this applies to self-injective algebras, hereditary algebras, canonical algebras, tilted and quasitilted algebras, piecewise hereditary algebras, etc. As opposed to investigating individuals from such "purified" classes, we propose to investigate whole emph{sequences of algebras} $(A_n)_{nin NN}$, where all $A_n$ follow the 'same' building principle and with the number $s(n)$ of simple $A_n$-modules converging with $n$ to $infty$. By way of example, let $A_n$ be the path algebra of the linear quiver $1up{x}2up{x}3 up{x} cdots up{x} n-1 up{x}n$ with $n$ vertices modulo all relations of the form $x^r=0$ (for a fixed integer $r$). Other such sequences arise from poset algebras (= algebras of fully commutative quivers) following a common building principle (concatenation of squares, diamonds, etc.). Typically, such sequences form a `trajectory' through the world of algebras as partitioned into traditionally studied classes like those mentioned before. Given such a sequence $(A_n)$, for small values of $n$ the algebra $A_n$ typically belongs to one of the standard classes (at least up to derived equivalence). By contrast for large values of $n$ the algebras $A_n$ typically belong to unknown territory.

As the present, still very preliminary analysis reveals, studying sequences of algebras, in particular reveals unexpected relations to singularity theory. (In the list of proposed participants we have therefore included experts from singularities.)

From the study of this problem we expect a reevaluation of (and some kind of measure for the importance of) existing classes of algebras, more importantly the development of new concepts in representation theory. But most of all, we expect to strengthen the links between representation theory and other subjects of mathematics.

Nilpotent operators

This topic is related to the first and the third theme, since the covering problem and the sequence problem are inbuilt. Its focus is the study of categories of invariant subspaces of nilpotent operators as studied by Ringel and Schmidmeier, giving rise a collection of difficult problems.

One tool studying this problem is the relationship between what Rouquier (in the inventiones paper on representation dimension) calls the derived categories of differential modules: The theory of such derived categories mirrors that of the usual derived category of complexes of modules, but forgets the grading. Thus we deal again with a covering situation which again has not yet been considered in detail anywhere! The categories of invariant subspaces of nilpotent operators form yet another class of categories which fit well into the scheme of being test models; also here, the covering theory plays an essential role - and we again encounter the challenge of dealing with sequences of problems.

The proposed Workshop intends to apply such tests to a number of directions by focussing on the following questions:

1. The representation theory of wild Kronecker algebras. The $r$-Kronecker algebra $K_r$ is the path algebra of the quiver

$ 1 defr{rightarrow} renewcommandarraystretch{0}begin{array}{c}

r r vdots r

end{array} 2$ ($r$ arrows). For each $r$ the algebra $K_r$ is hereditary, thus belongs to that class of algebras whose representation theory is best understood. For $r=1$ the representation type of $K_r$ is finite, for $r=2$ it is tame (and derived equivalent to the classification of coherent sheaves on the projective line), while for $rgeq 3$ the classification problem is wild, and generally thought to be hopeless.

We propose to attack this problem by means of emph{covering theory}. In the hands of Gabriel and his school, covering theory was very successfully applied to finite type and later with similar success extended to tame type by Dowbor-Skowronski. By now it is a standard tool to investigate tame algebras and their representations, but curiously a serious application of covering methods for wild type does not exist. For instance covering theory plays a key role in the recently completed classification of tame self-injective algebras (satisfying some natural extra conditions) by Skowronski and collaborators. Coming back to the proposed topic, we expect that attacking the representation theory of wild Kronecker algebras by covering theory will yield substantial new insight, provide a significant stimulus to further development, in particular introduce a fresh point of view in the study of representations for algebras of wild type in general. Because of the fundamental importance of the representation theory of the Kronecker quiver and its appearance in many situations in mathematics, we expect new lines of interactions to neighboring subjects as a natural byproduct of this investigation.

2. The unknown nature of Kerner's exotic space. For a wild hereditary $k$-algebra the indecomposable modules fall into three classes: the indecomposable preprojective (resp. preinjective) modules, each forming a single, hence countable, Auslander-Reiten component, and an infinite family of regular components, all of the same combinatorial shape $ZZAA_infty$. A fundamental, and very surprising, result of O. Kerner states that there are natural bijections (in general not unique) between the sets of regular components for any two (connected) wild hereditary algebras, thus giving rise to a unique universal parametrizing space $Omega$. The mathematical structure of this space is completely unknown, even conjecturally. $Omega$ should be thought of to be a space of quite exotic nature.

It is curious that the space $Omega$ occurs in other mathematical contexts as well, since $Omega$ also parametrizes

the set of regular components of a wild canonical algebra, or the set of Auslander-Reiten components of vector bundles on a weighted projective line of Euler characteristic $<0$, or --- again surprisingly --- the set of Auslander-Reiten components for the triangulated category $mathcal{D}^b_{sg}(R)$ of singularities for (graded) surface singularities of Brieskorn or Fuchsian type.

Our aim to put this as a topic for the intended workshop is twofold: first we want to complete the picture, showing that further algebras and categories yield the 'same' parametrizing space and second we want to collect (partial or full) evidence about its nature.

3. Sequences of algebras. Many classes of finite dimensional algebras have been extensively studied. For instance this applies to self-injective algebras, hereditary algebras, canonical algebras, tilted and quasitilted algebras, piecewise hereditary algebras, etc. As opposed to investigating individuals from such "purified" classes, we propose to investigate whole emph{sequences of algebras} $(A_n)_{nin NN}$, where all $A_n$ follow the 'same' building principle and with the number $s(n)$ of simple $A_n$-modules converging with $n$ to $infty$. By way of example, let $A_n$ be the path algebra of the linear quiver $1up{x}2up{x}3 up{x} cdots up{x} n-1 up{x}n$ with $n$ vertices modulo all relations of the form $x^r=0$ (for a fixed integer $r$). Other such sequences arise from poset algebras (= algebras of fully commutative quivers) following a common building principle (concatenation of squares, diamonds, etc.). Typically, such sequences form a `trajectory' through the world of algebras as partitioned into traditionally studied classes like those mentioned before. Given such a sequence $(A_n)$, for small values of $n$ the algebra $A_n$ typically belongs to one of the standard classes (at least up to derived equivalence). By contrast for large values of $n$ the algebras $A_n$ typically belong to unknown territory.

As the present, still very preliminary analysis reveals, studying sequences of algebras, in particular reveals unexpected relations to singularity theory. (In the list of proposed participants we have therefore included experts from singularities.)

From the study of this problem we expect a reevaluation of (and some kind of measure for the importance of) existing classes of algebras, more importantly the development of new concepts in representation theory. But most of all, we expect to strengthen the links between representation theory and other subjects of mathematics.

Nilpotent operators

This topic is related to the first and the third theme, since the covering problem and the sequence problem are inbuilt. Its focus is the study of categories of invariant subspaces of nilpotent operators as studied by Ringel and Schmidmeier, giving rise a collection of difficult problems.

One tool studying this problem is the relationship between what Rouquier (in the inventiones paper on representation dimension) calls the derived categories of differential modules: The theory of such derived categories mirrors that of the usual derived category of complexes of modules, but forgets the grading. Thus we deal again with a covering situation which again has not yet been considered in detail anywhere! The categories of invariant subspaces of nilpotent operators form yet another class of categories which fit well into the scheme of being test models; also here, the covering theory plays an essential role - and we again encounter the challenge of dealing with sequences of problems.