Finite dimensional algebras and quantum groups

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ContentsPrefaceNotational conventionsLeitfadenxiiixxiiixxvChapter 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 25Exercises and notes 34Part 1. Quivers and Their RepresentationsChapter 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 70Exercises and notes 74vii
viiiContentsChapter 2. Algebras with Frobenius morphisms 83§2.1. F q -structures on vector spaces 84§2.2. Algebras with Frobenius morphisms and Frobenius twists 86§2.3. F-stable A-modules 91§2.4. A construction of indecomposable F-stable modules 94§2.5. A functorial approach to the representation theory 98§2.6. Almost split sequences 105§2.7. Irreducible morphisms 112§2.8. Frobenius folding of almost split sequences 117Exercises and notes 121Chapter 3. Quivers with automorphisms 127§3.1. Quivers with automorphisms and valued quivers 128§3.2. Automorphisms of Dynkin and tame quivers 135§3.3. Modulated quivers and Auslander–Reiten quivers 140§3.4. Preprojective and preinjective components 145§3.5. Modulated quivers attached to quivers with automorphisms 150§3.6. Frobenius folding of Auslander–Reiten quivers 156§3.7. Finite dimensional algebras over a finite field 164§3.8. Representations of tame quivers with automorphisms 170Exercises and notes 174Part 2. Some Quantized AlgebrasChapter 4. Coxeter groups and Hecke algebras 183§4.1. Coxeter groups 184§4.2. An example: symmetric groups 193§4.3. Parabolic subgroups and affine Weyl groups 197§4.4. Hecke algebras 203§4.5. Hecke monoids 208§4.6. Counting with finite general linear groups 212§4.7. Integral Hecke algebras associated with GL n (q) 218Exercises and notes 223Chapter 5. Hopf algebras and universal enveloping algebras 229§5.1. Coalgebras, bialgebras, and Hopf algebras 230§5.2. Universal enveloping algebras and PBW bases 239
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