Noncommutative geometry and Cayley-smooth orders

569 Pages·2008·3.447 MB·English

by  Lieven Le Bruyn

Noncommutative Geometry
and Cayley-smooth Orders
© 2008 by Taylor & Francis Group, LLC
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© 2008 by Taylor & Francis Group, LLC
Noncommutative Geometry
and Cayley-smooth Orders
Lieven Le Bruyn
Universiteit Antwerpen
Belgium
Boca Raton London New York
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Library of Congress Cataloging‑in‑Publication Data
Le Bruyn, Lieven, 1958‑
Noncommutative geometry and Cayley‑smooth orders / Lieven Le Bruyn.
p. cm. ‑‑ (Pure and applied mathematics ; 290)
Includes bibliographical references and index.
ISBN 978‑1‑4200‑6422‑3 (alk. paper)
1. Noncommutative differential geometry. 2. Cayley‑Hamilton theorem. I.
Title. II. Series.
QC20.7.D52L4 2007
512’.55‑‑dc22 2007019964
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This book is dedicated to the women in my life
Simonne Stevens (1926-2004), Ann, Gitte & Bente
© 2008 by Taylor & Francis Group, LLC
Contents
Preface xiii
Introduction xv
I.1 Noncommutative algebra . . . . . . . . . . . . . . . . . . . . xvii
I.2 Noncommutative geometry . . . . . . . . . . . . . . . . . . . xxxv
I.3 Noncommutative desingularizations . . . . . . . . . . . . . . l
About the Author lxiii
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
ix
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x Contents
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
© 2008 by Taylor & Francis Group, LLC
Contents xi
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 K¨ahler structure . . . . . . . . . . . . . . . . . . . . . 482
8.6 Calogero particles . . . . . . . . . . . . . . . . . . . . . . . . 487
8.7 Coadjoint orbits . . . . . . . . . . . . . . . . . . . . . . . . . 493
8.8 Adelic Grassmannian . . . . . . . . . . . . . . . . . . . . . . 497
References 505
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