INTRODUCTION 9

of the commutativity of the function field in terms of the multiplicities (Section 4.3).

We have seen that several new and surprising phenomena occur when an ar-

bitrary base field is allowed. Along the way, we will point out several interesting

open problems. The following are particularly worth mentioning:

• Find graded factorial projective coordinate algebras for all weighted cases

(by a suitable method of inserting weights also into non-central prime

elements).

• Determine the ghost group in general. Describe the action of the

Auslander-Reiten translation on morphisms in general.

• The function field k(X) is always of finite dimension over its centre. Is

the square root of this dimension always the maximum of the multiplicity

function e? Describe each multiplicity e(x) in terms of the function field.

• Is it true that the completions R of the described graded factorial algebras

R are factorial again?

These notes are based on the author’s Habilitationsschrift with the title “As-

pects of hereditary representation theory over non-algebraically closed fields” ac-

cepted by the University of Paderborn in 2004. The present version includes further

recent results, in particular those concerning the multiplicities in Chapter 2.

We assume that the reader is familiar with the language of representation the-

ory of finite dimensional algebras. We refer to the books of Assem, Simson and

Skowro´ nski [3], of Auslander, Reiten and Smalø [5], and of Ringel [91].

Acknowledgements. It is a pleasure to express my gratitude to several people.

First I wish to thank Helmut Lenzing for many inspiring discussions on the subject.

Most of what I know about representation theory and weighted projective lines I

learned from him. He has always encouraged my interest in factoriality questions

in this context which started with my doctoral thesis and the generalization [55] of

a theorem of S. Mori [83]. For various helpful discussions and comments I would

like to thank Bill Crawley-Boevey, Idun Reiten and Claus Ringel. In particular,

some of the results concerning multiplicities were inspired by questions and com-

ments of Bill Crawley-Boevey and Claus Ringel. I also would like to thank Hagen

Meltzer for many discussions on several aspects of weighted projective lines and

exceptional curves. The section on the transitivity of the braid group action is a

short report on a joint work with him. I got the main idea for the definition of

an eﬃcient automorphism and thus for the verification of the graded factoriality in

full generality during a visit at the Mathematical Institute of the UNAM in Mexico

City when preparing a series of talks on the subject. I thank the colleagues of the

representation theory group there, in particular Michael Barot and Jos´ e Antonio de

la Pe˜ na, for their hospitality and for providing a stimulating working atmosphere.

For their useful advices and comments on various parts and versions of the manu-

script I thank Axel Boldt, Andrew Hubery and Henning Krause. For her love and

her patience I wish to thank my dear partner Gordana.