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
• 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 , of Auslander, Reiten and Smalø , and of Ringel .
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  of
a theorem of S. Mori . 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.