1. Introduction 3

linear group over C, representations of GL2(Fq), representations of

semidirect products, etc.

In Chapter 6, we give an introduction to the representation theory

of quivers (starting with the problem of the classification of configura-

tions of n subspaces in a vector space) and present a proof of Gabriel’s

theorem, which classifies quivers of finite type.

In Chapter 7, we give an introduction to category theory, in par-

ticular, abelian categories, and explain how such categories arise in

representation theory.

In Chapter 8, we give a brief introduction to homological algebra

and explain how it can be applied to categories of representations.

Finally, in Chapter 9 we give a short introduction to the repre-

sentation theory of finite dimensional algebras.

Besides, the book contains six historical interludes written by Dr.

Slava

Gerovitch.1 These interludes, written in an accessible and ab-

sorbing style, tell about the life and mathematical work of some of

the mathematicians who played a major role in the development of

modern algebra and representation theory: F. G. Frobenius, S. Lie,

W. Burnside, W. R. Hamilton, H. Weyl, S. Mac Lane, and S. Eilen-

berg. For more on the history of representation theory, we recommend

that the reader consult the references to the historical interludes, in

particular the excellent book [Cu].

Acknowledgments. This book arose from the lecture notes of

a representation theory course given by the first author to the re-

maining six authors in March 2004 within the framework of the Clay

Mathematics Institute Research Academy for high school students

and its extended version given by the first author to MIT undergrad-

uate mathematics students in the fall of 2008.

The authors are grateful to the Clay Mathematics Institute for

hosting the first version of this course. The first author is very in-

debted to Victor Ostrik for helping him prepare this course and thanks

1I

wish to thank Prof. Pavel Etingof and his co-authors for adding technical notes

to my historical monograph. While they have made a commendable effort at a concise

exposition, their notes, unfortunately, have grown in size and in the end occupied

a better part of this volume. I hope the reader will forgive this preponderance of

technicalities in what, in essence, is a history book. — S. Gerovitch.