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An Introduction to Central Simple Algebras and Their Applications to Wireless Communication Mathematical Surveys and Monographs Volume 191 An Introduction to Central Simple Algebras and Their Applications to Wireless Communication Grégory Berhuy Frédérique Oggier American Mathematical Society http://dx.doi.org/10.1090/surv/191 An Introduction to Central Simple Algebras and Their Applications to Wireless Communication Mathematical Surveys and Monographs Volume 191 An Introduction to Central Simple Algebras and Their Applications to Wireless Communication Grégory Berhuy Frédérique Oggier American Mathematical Society Providence, Rhode Island EDITORIAL COMMITTEE Ralph L. Cohen, Chair Benjamin Sudakov Robert Guralnick MichaelI.Weinstein MichaelA.Singer 2010 Mathematics Subject Classification. Primary 12E15; Secondary 11T71, 16W10. For additional information and updates on this book, visit www.ams.org/bookpages/surv-191 Library of Congress Cataloging-in-Publication Data Berhuy, Gr´egory. An introduction to central simple algebras and their applications to wireless communications /Gr´egory Berhuy, Fr´ed´erique Oggier. pages cm. – (Mathematical surveys and monographs ; volume 191) Includes bibliographical references and index. ISBN 978-0-8218-4937-8 (alk. paper) 1. Division algebras. 2. Skew fields. I. Oggier, Fr´ed´erique. II. Title. QA247.45.B47 2013 512.3–dc23 2013009629 Copying and reprinting. Individual readers of this publication, and nonprofit libraries acting for them, are permitted to make fair use of the material, such as to copy a chapter for use in teaching or research. Permission is granted to quote brief passages from this publication in reviews, provided the customary acknowledgment of the source is given. Republication, systematic copying, or multiple reproduction of any material in this publication is permitted only under license from the American Mathematical Society. Requests for such permission should be addressed to the Acquisitions Department, American Mathematical Society, 201 Charles Street, Providence, Rhode Island 02904-2294 USA. Requests can also be made by e-mail to [email protected]. c 2013 by the American Mathematical Society. All rights reserved. The American Mathematical Society retains all rights except those granted to the United States Government. Printed in the United States of America. ∞ The paper used in this book is acid-free and falls within the guidelines established to ensure permanence and durability. Visit the AMS home page at http://www.ams.org/ 10987654321 181716151413 Contents Foreword.................................................................... vii Introduction................................................................. 1 ChapterI. Centralsimplealgebras.......................................... 3 I.1. Preliminaries on k-algebras.......................................... 3 I.2. Centralsimplealgebras: thebasics................................... 7 I.3. Introducingspace-timecoding....................................... 11 Exercises.................................................................. 18 ChapterII. Quaternionalgebras............................................ 21 II.1. Propertiesofquaternionalgebras................................... 21 II.2. Hamiltonquaternions............................................... 27 II.3. Quaternionalgebrasbasedcodes.................................... 28 Exercises.................................................................. 30 Chapter III. Fundamental results on central simple algebras. ................ 31 III.1. Operationsoncentralsimplealgebras.............................. 31 III.2. Simplemodules.................................................... 35 III.3. Skolem-Noether’stheorem.......................................... 43 III.4. Wedderburn’stheorem............................................. 45 III.5. Thecentralizertheorem............................................ 47 Exercises.................................................................. 50 ChapterIV. Splittingfieldsofcentralsimplealgebras....................... 53 IV.1. Splittingfields..................................................... 53 IV.2. Thereducedcharacteristicpolynomial.............................. 60 IV.3. Theminimumdeterminantofa code............................... 68 Exercises.................................................................. 76 ChapterV. TheBrauergroupofa field..................................... 79 V.1. DefinitionoftheBrauergroup...................................... 79 V.2. Brauerequivalenceandbimodules................................... 82 V.3. Indexandexponent................................................. 91 Exercises.................................................................. 98 ChapterVI. Crossedproducts.............................................. 101 VI.1. Definitionofcrossedproducts...................................... 101 VI.2. Somepropertiesofcrossedproducts................................108 VI.3. Shapingandcrossedproductsbasedcodes..........................118 Exercises..................................................................126 v vi CONTENTS ChapterVII. Cyclicalgebras................................................129 VII.1. Cyclicalgebras....................................................129 VII.2. Centralsimplealgebrasoverlocalfields........................... 137 VII.3. Centralsimplealgebrasovernumberfields.........................139 VII.4. Cyclicalgebrasofprimedegreeovernumberfields.................141 VII.5. Examples......................................................... 144 VII.6. Cyclicalgebrasandperfectcodes..................................150 VII.7. Optimalityofsomeperfectcodes.................................. 156 Exercises..................................................................163 ChapterVIII. Centralsimplealgebrasofdegree4...........................165 VIII.1. A theoremofAlbert..............................................165 VIII.2. Structureofcentralsimplealgebrasofdegree4................... 168 VIII.3. Albert’sTheorem................................................ 176 VIII.4. Codesoverbiquadraticcrossedproducts..........................178 Exercises..................................................................187 ChapterIX. Centralsimplealgebraswithunitaryinvolutions............... 189 IX.1. Basicconcepts. ....................................................189 IX.2. Thecorestrictionalgebra. ......................................... 191 IX.3. Existenceofunitaryinvolutions. .................................. 198 IX.4. Unitaryinvolutionsoncrossedproducts. .......................... 203 IX.5. Unitaryspace-timecoding..........................................209 Exercises..................................................................228 AppendixA. Tensorproducts...............................................231 A.1. Tensorproductofvectorspaces..................................... 231 A.2. Basicpropertiesofthetensorproduct...............................235 A.3. Tensor product of k-algebras........................................242 AppendixB. A glimpseofnumbertheory...................................249 B.1. Absolutevalues..................................................... 249 B.2. Factorizationofidealsinnumberfields..............................253 B.3. Absolutevaluesonnumberfieldsandcompletion....................262 AppendixC. Complexideallattices.........................................265 C.1. Generalitiesonhermitianlattices....................................265 C.2. Complexideallattices...............................................266 Bibliography.................................................................271 Index........................................................................ 275 Foreword Mathematics continually surprises and delights us with how useful its most abstract branches turn out to be in the real world. Indeed, physicist Eugene Wigner’s mem- orable phrase1 “The unreasonable effectiveness of mathematics” captures a critical aspect of this utility. Abstract mathematical ideas often prove to be useful in rather “unreasonable” situations: places where one, a priori, would not expect them at all! For instance, no one who was not actually following the theoretical explorations in multi-antenna wireless communication of the late 1990s would have predicted that division algebras would turn out to be vital in the deployment of multi-antenna communication. Yet, once performance criteria for space-time codes (as coding schemes for multi-antenna environments are called) were developed and phrased as a problem of design of matrices, it was completely natural that division algebras should arise as a solution of the design problem. The fundamental performance criterion ask for n × n matrices Mi such that the difference of any two of the Mi is of full rank. To anyone who has worked with division algebras, the solution simply leaps out: any division algebra of index n embeds into the n × n matrices over a suitable field, and the matrices arising from the embedding naturally satisfy this criterion. But there is more. Not only did division algebras turn out to be the most natural context in which to solve the fundamental design problem above, they also proved to be the correct objects to satisfy various other performance criteria that were devel- oped. For instance, a second, and critical, performance criterion called the coding gain criterion turned out to be naturally satisfied by considering division algebras over number fields and using natural Z-orders within them that arise from rings of integers of maximal subfields. Other criteria (for instance
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