Springer Undergraduate Mathematics Series
Editorial Board
M. A. J. Chaplain
University of St. Andrews, UK
A. MacIntyre
Queen Mary University of London, UK
S. Scott
King’s College London, UK
N. Snashall
University of Leicester, UK
E. Süli
University of Oxford, UK
M. R. Tehranchi
University of Cambridge, UK
J. F. Toland
University of Bath, UK

More information about this series at http://​www.​springer.​com/​series/​3423

Karin Erdmann and Thorsten Holm
Algebras and Representation Theory
Introduction

Representation theory is a beautiful subject which has numerous applications in mathematics and beyond. Roughly speaking, representation theory investigates how algebraic systems can act on vector spaces. When the vector spaces are finite-dimensional this allows one to explicitly express the elements of the algebraic system by matrices, and hence one can exploit basic linear algebra to study abstract algebraic systems. For example, one can study symmetry via group actions, but more generally one can also study processes which cannot be reversed. Algebras and their representations provide a natural framework for this. The idea of letting algebraic systems act on vector spaces is so general that there are many variations. A large part of these fall under the heading of representation theory of associative algebras, and this is the main focus of this text.

Examples of associative algebras which already appear in basic linear algebra are the spaces of n  ×  n -matrices with coefficients in some field K , with the usual matrix operations. Another example is provided by polynomials over some field, but there are many more. In general, roughly speaking, an associative algebra A is a ring which also is a vector space over some field K such that scalars commute with all elements of A . We start by introducing algebras and basic concepts, and describe many examples. In particular, we discuss group algebras, division algebras, and path algebras of quivers. Next, we introduce modules and representations of algebras and study standard concepts, such as submodules, factor modules and module homomorphisms.

A module is simple (or irreducible) if it does not have any submodules, except zero and the whole space. The first part of the text is motivated by simple modules. They can be seen as building blocks for arbitrary modules, this is made precise by the Jordan–Hölder theorem. It is therefore a fundamental problem to understand all simple modules of an algebra. The next question is then how the simple modules can be combined to form new modules. For some algebras, every module is a direct sum of simple modules, and in this case, the algebra is called semisimple. We study these in detail. In addition, we introduce the Jacobson radical of an algebra, which, roughly speaking, measures how far an algebra is away from being semisimple. The Artin–Wedderburn theorem completely classifies semisimple algebras. Given an arbitrary algebra, in general it is difficult to decide whether or not it is semisimple. However, when the algebra is the group algebra of a finite group G , Maschke’s theorem answers this, namely the group algebra KG is semisimple if and only the characteristic of the field K does not divide the order of the group. We give a proof, and we discuss some applications.

If an algebra is not semisimple, one has to understand indecomposable modules instead of just simple modules. The second half of the text focusses on these. Any finite-dimensional module is a direct sum of indecomposable modules. Even more, such a direct sum decomposition is essentially unique, this is known as the Krull–Schmidt theorem. It shows that it is enough to understand indecomposable modules of an algebra. This suggests the definition of the representation type of an algebra. This is said to be finite if the algebra has only finitely many indecomposable modules, and is infinite otherwise. In general it is difficult to determine which algebras have finite representation type. However for group algebras of finite groups, there is a nice answer which we present.

The rest of the book studies quivers and path algebras of quivers. The main goal is to classify quivers whose path algebras have finite representation type. This is Gabriel’s theorem, proved in the 1970s, which has led to a wide range of new directions, in algebra and also in geometry and other parts of mathematics. Gabriel proved that a path algebra KQ of a quiver Q has finite representation type if and only if the underlying graph of Q is the disjoint union of Dynkin diagrams of types A , D or E . In particular, this theorem shows that the representation type of KQ is independent of the field K and is determined entirely by the underlying graph of Q . Our aim is to give an elementary account of Gabriel’s theorem, and this is done in several chapters. We introduce representations of quivers; they are the same as modules for the path algebra of the quiver, but working with representations has additional combinatorial information. We devote one chapter to the description of the graphs relevant for the proof of Gabriel’s theorem, and to the development of further tools related to these graphs. Returning to representations, we introduce reflections of quivers and of representations, which are crucial to show that the representation type does not depend on the orientation of arrows. Combining the various tools allows us to prove Gabriel’s Theorem, for arbitrary fields.

This text is an extended version of third year undergraduate courses which we gave at Oxford, and at Hannover. The aim is to give an elementary introduction, we assume knowledge of results from linear algebra, and on basic properties of rings and groups. Apart from this, we have tried to make the text self-contained. We have included a large number of examples, and also exercises. We are convinced that they are essential for the understanding of new mathematical concepts. In each section, we give sample solutions for some of the exercises, which can be found in the appendix. We hope that this will help to make the text also suitable for independent self-study.

Karin Erdmann
Thorsten Holm
Oxford, UKHannover, Germany
2018

Contents

A Induced Modules for Group Algebras265
B Solutions to Selected Exercises271
Index297