Contents
Preface xiii
Notational conventions xxiii
Leitfaden xxv
Chapter 0. Getting started 1
§0.1. Cartan matrices and their two realizations 1
§0.2. Free algebras and presentations with generators and relations 6
§0.3. Examples: the realization problem 12
§0.4. Counting over finite fields: Gaussian polynomials 17
§0.5. Canonical bases: the matrix construction 22
§0.6. Finite dimensional semisimple Lie algebras 25
Exercises and notes 34
Part 1. Quivers and Their Representations
Chapter 1. Representations of quivers 43
§1.1. Quivers and their representations 44
§1.2. Euler forms, Cartan matrices, and the classification of quivers 49
§1.3. Weyl groups and root systems 55
§1.4. Bernstein-Gelfand-Ponomarev reflection functors 60
§1.5. Gabriel's theorem 65
§1.6. Representation varieties and generic extensions 70
Exercises and notes 74
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