Tame Algebras and Integral Quadratic Forms
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1099
Claus Michael Ringel
Tame Algebras and Integral Quadratic Forms
Spri nger-Verlag Berlin Heidelberg New York Tokyo 1984
Author
Claus Michael Ringel Fakultat fOr Mathematik, Universitat Bielefeld Postfach 8640, 4800 Bielefeld, Federal Republic of Germany
AMS Subject Classification (1980): 05C20, 06A10, 10B05, 15A30, 16A46, 16A48, 16A62, 16A64, 16-02, 18E10 ISBN 3-540-13905-2 Springer-Verlag Berlin Heidelberg New York Tokyo ISBN 0-387-13905-2 Springer-Verlag New York Heidelberg Berlin Tokyo
This work IS subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically those of translation, reprinting, re-use of illustrations, broadcasting, reproduction by photocopying machine or similar means, and storage in data banks. Under § 54 of the German Copyright Law where copies are made for other than private use, a fee is payable to "Verwertungsgesellschaft Wort", Munich.
© by Springer-Verlag Berlin Heidelberg 1984 Printed in Germany Printing and binding: Beltz Offsetdruck, Hemsbach/Bergstr. 2146/3140-543210
TAB LEO F
CON TEN T S
Introduction Integral quadratic forms 1.0 1.1
Two theorems of Ovsienko Roots of an integral quadratic form derivatives
DiX
of
X
and the partial
2
X
7
1.2
Dynkin graphs and Euclidean graphs
1.3
Graphical forms
II
1.4
Reduction to graphical forms
14
1.5
The quadratic forms occurring in tables I and 2
18
1.6
Maximal sincere positive roots of graphical forms with a unique exceptional index
22
1.7
Completeness of table
24
1.8
Proof of theorem 2
30
1.9
Completeness of table 2
31
1.10
The extended quadratic form of a finite partially ordered set
35
2
Quivers, module categories, subspace categories (notation, results, some proofs)
41 42
2.1
Quivers and translation quivers
2.2
Krull-Schmidt k-categories
52
2.3
Exact categories
59
2.4
Modules over (finite dimensional) algebras
66
2.5
Subspace categories and one-point extensions of algebras
82
2.6
Subspace categories of directed vectorspace categories and representations of partially ordered sets
97
Construction of stable separating tubular families
113
3.1
Separating tubular families
113
3.2
Example: Kronecker modules
122
3
3.3
Wing modules
127
3.4
The main theorem
130
3.5
The operation of
3.6
Tame hereditary algebras
153
3.7
Examples: The canonical algebras
161
¢A
on
Ko(A)
149
IV
4
Tilting functors and tubular extensions (notation , results, some proofs)
167
4.1
Tilting modules
167
4.2
Tilted algebras
179
4.3
Concealed algebras
192
4.4
Branches
202
4.5
Ray modules
214
4.6
Tubes
221
4.7
Tubular extensions
230
4.8
Examples: Canonical tubular extensions of canonical algebras
235
4.9
Domestic tubular extensions of tame concealed algebras
241
4.10
The critical directed vectorspace categories and their tubular extensions
254
5
Tubular algebras
268
5.1
Ko(A)
269
5.2
The structure of the module category of a tubular algebra
273
for a tubular and cotubular algebra
5.3
Some
