23 mar. 2015

Tame Algebras and Integral Quadratic Forms - Claus Michael Ringel





TABLE OF CONTENTS

Introduction

1 Integral quadratic forms

1.0 Two theorems of Ovsienko
1.2 Roots of an integral quadratic form × and the partial derivatives DiX of X
1.2 Dynkin graphs and Euclidean graphs
1.3 Graphical forms
1.4 Reduction to graphical forms
1.5 The quadratic forms occurring in tables I and 2
1.6 Maximal sincere positive roots of graphical forms with a unique exceptional index
1.7 Completeness of table I
1.8 Proof of theorem 2
1.9 Completeness of table 2
1.I0 The extended quadratic form of a finite partially ordered set

2 Quivers, module categories, subspaee categories (notation, results, some proofs)

2.1 Quivers and translation quivers
2.2 Krull-Schmidt k-categories
2.3 Exact categories
2.4 Modules over (finite dimensional) algebras
2.5 Subspace categories and one-point extensions of algebras
2.6 Subspaee categories of directed vectorspace categories and representations of partially ordered sets
3 Construction of stable separatin$ tubular families

3.1 Separating tubular families
3.2 Example: Kronecker modules
3.3 Wing modules
3.4 The main theorem
3.5 The operation of ~A on Ko(A)
3.6 Tame hereditary algebras
3.7 Examples: The canonical algebras

4 Tilting functors and tubular extensions (notation , results, some proofs)

4.1 Tilting modules
4.2 Tilted algebras
4.3 Concealed algebras
4.4 Branches
4.5 Ray modules
4.6 Tubes
4.7 Tubular extensions
4.8 Examples: Canonical tubular extensions of canonical algebras
4.9 Domestic tubular extensions of tame concealed algebras
4.10 The critical directed vectorspaee categories and their tubular extensions

5 Tubular algebras

5.1 Ko(A) for a tubular and cotubular algebra
5.2 The structure of the module category of a tubular algebra
5.3 Some further properties of stable tubular families
5.4 Shrinking functors
5.5 Tilting modules for tubular algebras
5.6 Self-reproduction of tubular families
5.7 The general case
5.8 Tubular vectorspace categories

6. Directed algebras

6.1 The orbit quiver of a sincere directed algebra
6.2 Sincere directing wing modules of Dynkin type
6.3 The large sincere directed algebras
6.4 Auslander-Reiten sequences with four middle terms
6.5 The inductive construction of sincere directed algebras

Appendix

A.I The periodic additive functions on ZA, with A Euclidean

A.2 The frames of the tame concealed algebras

A.3 The tubular patterns

Bibliography
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