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Noncommutative Geometry and Cayley-smooth Orders © 2008 by Taylor & Francis Group, LLC PURE AND APPLIED MATHEMATICS A Program of Monographs, Textbooks, and Lecture Notes EXECUTIVE EDITORS Earl J. Taft Zuhair Nashed Rutgers University University of Central Florida Piscataway, New Jersey Orlando, Florida EDITORIAL BOARD M. S. Baouendi Anil Nerode University of California, Cornell University San Diego Freddy van Oystaeyen Jane Cronin University of Antwerp, Rutgers University Belgium Jack K. Hale Donald Passman Georgia Institute of Technology University of Wisconsin, Madison S. Kobayashi University of California, Fred S. Roberts Berkeley Rutgers University Marvin Marcus David L. Russell University of California, Virginia Polytechnic Institute Santa Barbara and State University W. S. Massey Walter Schempp Yale University Universität Siegen © 2008 by Taylor & Francis Group, LLC MONOGRAPHS AND TEXTBOOKS IN PURE AND APPLIED MATHEMATICS Recent Titles J. Galambos and I. 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Johnson, Vikram Jha, and Mauro Biliotti, Handbook of Finite Translation Planes (2007) Lieven Le Bruyn, Noncommutative Geometry and Cayley-smooth Orders (2008) © 2008 by Taylor & Francis Group, LLC Noncommutative Geometry and Cayley-smooth Orders Lieven Le Bruyn Universiteit Antwerpen Belgium Boca Raton London New York Chapman & Hall/CRC is an imprint of the Taylor & Francis Group, an informa business © 2008 by Taylor & Francis Group, LLC Chapman & Hall/CRC Taylor & Francis Group 6000 Broken Sound Parkway NW, Suite 300 Boca Raton, FL 33487‑2742 © 2008 by Taylor & Francis Group, LLC Chapman & Hall/CRC is an imprint of Taylor & Francis Group, an Informa business No claim to original U.S. Government works Printed in the United States of America on acid‑free paper 10 9 8 7 6 5 4 3 2 1 International Standard Book Number‑13: 978‑1‑4200‑6422‑3 (Hardcover) This book contains information obtained from authentic and highly regarded sources. Reprinted material is quoted with permission, and sources are indicated. A wide variety of references are listed. Reasonable efforts have been made to publish reliable data and information, but the author and the publisher cannot assume responsibility for the validity of all materials or for the conse‑ quences of their use. No part of this book may be reprinted, reproduced, transmitted, or utilized in any form by any electronic, mechanical, or other means, now known or hereafter invented, including photocopying, microfilming, and recording, or in any information storage or retrieval system, without written permission from the publishers. For permission to photocopy or use material electronically from this work, please access www. copyright.com (http://www.copyright.com/) or contact the Copyright Clearance Center, Inc. (CCC) 222 Rosewood Drive, Danvers, MA 01923, 978‑750‑8400. CCC is a not‑for‑profit organization that provides licenses and registration for a variety of users. For organizations that have been granted a photocopy license by the CCC, a separate system of payment has been arranged. Trademark Notice: Product or corporate names may be trademarks or registered trademarks, and are used only for identification and explanation without intent to infringe. 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 Visit the Taylor & Francis Web site at http://www.taylorandfrancis.com and the CRC Press Web site at http://www.crcpress.com © 2008 by Taylor & Francis Group, LLC 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 © 2008 by Taylor & Francis Group, LLC 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 Kähler structure . . . . . . . . . . . . . . . . . . . . . 482 8.6 Calogero particles . . . . . . . . . . . . . . . . . . . . . . . . 487 8.7 Coadjoint orbits . . . . . . . . . . . . . . . . . . . . . . . . . 493 8.8 Adelic Grassmannian . . . . . . . . . . . . . . . . . . . . . . 497 References 505 © 2008 by Taylor & Francis Group, LLC