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Introduction to the representation theory of algebras Michael Barot October 20, 2011
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Preface The aim of this notes is to give a brief and elementary introduction to the representation theory of finite-dimensional algebras. The notes had its origin in a undergraduate course I gave in two occasions at Universidad Nacional Auton´ oma de M´exico. The plan of the course was to try to cope with two competing demands: to expect as little as possible and to reach as much as possible: to expect only linear algebra as background and yet to make way to substantial and central ideas and results during its progress. Therefore some crucial decisions were necessary. We opted for the model case rather than the most general situation, to the most illustrating example rather than the most extravagant one. We sought a guideline through this vast field which conducts to as many important notions, techniques and questions as possible in the limited space of a one-semester course. So, it is a book written from a specific point of view and the title should really be Introduction to the theory of algebras, which are finite dimensional over some algebraically closed field. The book starts with the most difficult chapter in front: matrix problems. Conceptually there is little to understand in that chapter but it requires a considerable effort from the reader to follow the argumentation within. However, this chapter is central: it prepares all the main examples which later will guide through the rest. In the following two chapters we consider the main languages of representation theory. Since there are several competing languages in representation theory, a considerable amount of our effort is directed towards mastering and combining all of them. As you will see each of these languages has its own advantages and it therefore enables the reader not only to consult the majority of all research articles in the field, it also enriches the way we may thick about the notions themselves. The rest of the book is devoted to gain structural insight into the categories
vi of modules of a given algebra. In the chapter about module categories some older results are proved, whereas in the next four chapters some newer insights are transmitted. The last chapter is more of an open minded collection concerning what can be said about the building bricks of the module categories, which are called indecomposables with respect to certain invariants, called dimension vectors, under the action of isomorphisms.
Contents 1 Matrix problems
1
1.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1
1.2
The two subspace problem . . . . . . . . . . . . . . . . . . . .
3
1.3
Decomposition into indecomposables . . . . . . . . . . . . . .
5
1.4
The Kronecker problem . . . . . . . . . . . . . . . . . . . . .
7
1.5
The three Kronecker problem . . . . . . . . . . . . . . . . . .
12
1.6
Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
14
2 Representations of quivers
17
2.1
Quivers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
17
2.2
Representations . . . . . . . . . . . . . . . . . . . . . . . . . .
19
2.3
Categories and functors . . . . . . . . . . . . . . . . . . . . .
22
2.4
The path category . . . . . . . . . . . . . . . . . . . . . . . .
26
2.5
Equivalence of categories . . . . . . . . . . . . . . . . . . . . .
29
2.6
A new example . . . . . . . . . . . . . . . . . . . . . . . . . .
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3 Algebras
37
3.1
Def