Noncommutative geometry (web draft, version 3)

Le Bruyn L.

Noncommutative geometry is the study of families of (commutative) algebraicvarieties (with specified connecting morphisms) which are locally controlledby noncommutative algebras.

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Table of contents :
alg @n……Page 5
1.1. Conjugacy classes of matrices……Page 7
1.2. Simultaneous conjugacy classes…….Page 15
1.3. Matrix invariants and necklaces……Page 18
1.4. The trace algebra…….Page 23
1.5. The symmetric group…….Page 27
1.6. Necklace relations…….Page 29
1.7. Trace relations…….Page 35
1.8. Cayley-Hamilton algebras…….Page 39
References…….Page 43
2.1. Representation spaces…….Page 45
2.2. Some algebraic geometry…….Page 47
2.3. The Hilbert criterium…….Page 51
2.4. Semisimple modules……Page 54
2.5. Some invariant theory…….Page 58
2.6. Geometric reconstruction……Page 64
2.7. The Gerstenhaber-Hesselink theorem…….Page 68
2.8. The real moment map…….Page 76
top @n……Page 81
3.1. Etale topology…….Page 85
3.2. Central simple algebras……Page 91
3.3. Quiver orders…….Page 94
3.4. Simple roots…….Page 105
3.5. Spectral sequences……Page 110
3.6. Tsen and Tate fields……Page 113
3.7. Coniveau spectral sequence……Page 116
3.8. The Artin-Mumford exact sequence……Page 119
References…….Page 124
4.1. C slices…….Page 125
4.2. Normal spaces…….Page 126
4.3. Knop-Luna slices…….Page 132
4.4. Smoothness…….Page 136
4.5. Local structure…….Page 141
4.6. A-algebras…….Page 147
4.7. Indecomposable roots…….Page 150
4.8. Canonical decomposition…….Page 157
loc @n……Page 166
5.1. Type stratification…….Page 171
5.2. Cayley-smooth locus…….Page 176
5.3. Local classification…….Page 183
5.4. Low dimensional orders…….Page 185
5.5. Noncommutative smooth surfaces…….Page 192
5.6. Complex moment map…….Page 197
5.7. Preprojective algebras…….Page 201
5.8. Quantum groups…….Page 203
References…….Page 209
6.1. Cornering matrices…….Page 211
6.2. General subrepresentations…….Page 215
6.3. Semistable representations…….Page 219
6.4. Optimal corners…….Page 225
6.5. Hesselink stratification…….Page 228
6.6. Cornering quiver representations…….Page 234
6.7. Simultaneous conjugacy classes…….Page 239
6.8. Representation fibers…….Page 245
geo @n……Page 251
7.1. Formal structure…….Page 255
7.2. Semi invariants…….Page 260
7.3. Universal localization…….Page 268
7.4. Compact manifolds…….Page 274
7.5. Differential forms…….Page 281
7.6. deRham cohomology…….Page 292
7.7. Symplectic structure…….Page 298
7.8. Necklace Lie algebras…….Page 303
References…….Page 306
8.1. Moment maps…….Page 307
8.2. Dynamical systems…….Page 310
8.3. Deformed preprojective algebras…….Page 316
8.4. Hilbert schemes…….Page 319
8.5. Hyper Kähler structure…….Page 329
8.6. Calogero particles…….Page 333
8.7. Coadjoint orbits…….Page 337
8.8. Adelic Grassmannian…….Page 340
Bibliography……Page 344
Index……Page 345