Noncommutative geometry and Cayley-smooth orders

Noncommutative geometry and Cayley-smooth orders [electronic resource] / Lieven Le Bruyn.

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Bibliographic Details
Online Access: Full Text (via Springer)
Main Author: Le Bruyn, Lieven, 1958-
Format: Electronic eBook
Language:English
Published: Boca Raton : Chapman & Hall/CRC, ©2008.
Series:Monographs and textbooks in pure and applied mathematics ; 290.
Subjects:
Noncommutative differential geometry.
Cayley-Hamilton theorem.
Table of Contents:
  • I.1 Noncommutative algebra xvii
  • I.2 Noncommutative geometry xxxv
  • I.3 Noncommutative desingularizations 1
  • 1 Cayley-Hamilton Algebras 1
  • 1.1 Conjugacy classes of matrices 1
  • 1.2 Simultaneous conjugacy classes 14
  • 1.3 Matrix invariants and necklaces 18
  • 1.4 The trace algebra 25
  • 1.5 The symmetric group 30
  • 1.6 Necklace relations 33
  • 1.7 Trace relations 42
  • 1.8 Cayley-Hamilton algebras 47
  • 2 Reconstructing Algebras 55
  • 2.1 Representation schemes 55
  • 2.2 Some algebraic geometry 58
  • 2.3 The Hilbert criterium 64
  • 2.4 Semisimple modules 69
  • 2.5 Some invariant theory 75
  • 2.6 Geometric reconstruction 82
  • 2.7 The Gerstenhaber-Hesselink theorem 89
  • 2.8 The real moment map 99
  • 3 Etale Technology 109
  • 3.1 Etale topology 109
  • 3.2 Central simple algebras 119
  • 3.3 Spectral sequences 123
  • 3.4 Tsen and Tate fields 127
  • 3.5 Coniveau spectral sequence 132
  • 3.6 The Artin-Mumford exact sequence 136
  • 3.7 Normal spaces 145
  • 3.8 Knop-Luna slices 155
  • 4 Quiver Technology 163
  • 4.1 Smoothness 163
  • 4.2 Local structure 174
  • 4.3 Quiver orders 185
  • 4.4 Simple roots 197
  • 4.5 Indecomposable roots 203
  • 4.6 Canonical decomposition 213
  • 4.7 General subrepresentations 226
  • 4.8 Semistable representations 231
  • 5 Semisimple Representations 241
  • 5.1 Representation types 241
  • 5.2 Cayley-smooth locus 249
  • 5.3 Reduction steps 258
  • 5.4 Curves and surfaces 271
  • 5.5 Complex moment map 285
  • 5.6 Preprojective algebras 291
  • 5.7 Central smooth locus 294
  • 5.8 Central singularities 304
  • 6 Nilpotent Representations 315
  • 6.1 Cornering matrices 315
  • 6.2 Optimal corners 322
  • 6.3 Hesselink stratification 326
  • 6.4 Cornering quiver representations 335
  • 6.5 Simultaneous conjugacy classes 342
  • 6.6 Representation fibers 350
  • 6.7 Brauer-Severi varieties 362
  • 6.8 Brauer-Severi fibers 368
  • 7 Noncommutative Manifolds 377
  • 7.1 Formal structure 377
  • 7.2 Semi-invariants 385
  • 7.3 Universal localization 396
  • 7.4 Compact manifolds 404
  • 7.5 Differential forms 413
  • 7.6 deRham cohomology 429
  • 7.7 Symplectic structure 438
  • 7.8 Necklace Lie algebras 445
  • 8 Moduli Spaces 451
  • 8.1 Moment maps 452
  • 8.2 Dynamical systems 456
  • 8.3 Deformed preprojective algebras 464
  • 8.4 Hilbert schemes 469
  • 8.5 Hyper Kahler structure 482
  • 8.6 Calogero particles 487
  • 8.7 Coadjoint orbits 493
  • 8.8 Adelic Grassmannian 497.

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