vi Contents §5.27. Representations of semidirect products 142 Chapter 6. Quiver representations 145 §6.1. Problems 145 §6.2. Indecomposable representations of the quivers A1,A2,A3 150 §6.3. Indecomposable representations of the quiver D4 154 §6.4. Roots 160 §6.5. Gabriel’s theorem 163 §6.6. Reflection functors 164 §6.7. Coxeter elements 169 §6.8. Proof of Gabriel’s theorem 170 §6.9. Problems 173 Chapter 7. Introduction to categories 177 §7.1. The definition of a category 177 §7.2. Functors 179 §7.3. Morphisms of functors 181 §7.4. Equivalence of categories 182 §7.5. Representable functors 183 §7.6. Adjoint functors 184 §7.7. Abelian categories 186 §7.8. Complexes and cohomology 187 §7.9. Exact functors 190 §7.10. Historical interlude: Eilenberg, Mac Lane, and “general abstract nonsense” 192 Chapter 8. Homological algebra 201 §8.1. Projective and injective modules 201 §8.2. Tor and Ext functors 203 Chapter 9. Structure of finite dimensional algebras 209 §9.1. Lifting of idempotents 209 §9.2. Projective covers 210
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