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Mathematical Surveys
and Monographs
Volume 191
American Mathematical Society
An Introduction to Central Simple Algebras and Their Applications to Wireless Communication
Grégory Berhuy Frédérique Oggier
An Introduction to Central Simple Algebras and Their Applications to Wireless Communication
http://dx.doi.org/10.1090/surv/191
Mathematical Surveys
and Monographs
Volume 191
An Introduction to Central Simple Algebras and Their Applications to Wireless Communication
Grégory Berhuy Frédérique Oggier
American Mathematical SocietyProvidence, Rhode Island
EDITORIAL COMMITTEE
Ralph L. Cohen, ChairRobert GuralnickMichael A. Singer
Benjamin SudakovMichael I. Weinstein
2010 Mathematics Subject Classification. Primary 12E15; Secondary 11T71, 16W10.
For additional information and updates on this book, visitwww.ams.org/bookpages/surv-191
Library of Congress Cataloging-in-Publication Data
Berhuy, Gregory.An introduction to central simple algebras and their applications to wireless communications
/ Gregory Berhuy, Frederique 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, Frederique. II. Title.
QA247.45.B47 2013512′.3–dc23 2013009629
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10 9 8 7 6 5 4 3 2 1 18 17 16 15 14 13
Contents
Foreword . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
Chapter I. Central simple algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3I.1. Preliminaries on k-algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3I.2. Central simple algebras: the basics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7I.3. Introducing space-time coding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
Chapter II. Quaternion algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21II.1. Properties of quaternion algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21II.2. Hamilton quaternions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27II.3. Quaternion algebras based codes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
Chapter III. Fundamental results on central simple algebras. . . . . . . . . . . . . . . . . 31III.1. Operations on central simple algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31III.2. Simple modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35III.3. Skolem-Noether’s theorem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43III.4. Wedderburn’s theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45III.5. The centralizer theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
Chapter IV. Splitting fields of central simple algebras . . . . . . . . . . . . . . . . . . . . . . . 53IV.1. Splitting fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53IV.2. The reduced characteristic polynomial . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60IV.3. The minimum determinant of a code . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
Chapter V. The Brauer group of a field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79V.1. Definition of the Brauer group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79V.2. Brauer equivalence and bimodules. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82V.3. Index and exponent . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
Chapter VI. Crossed products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101VI.1. Definition of crossed products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101VI.2. Some properties of crossed products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108VI.3. Shaping and crossed products based codes. . . . . . . . . . . . . . . . . . . . . . . . . . 118Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126
v
vi CONTENTS
Chapter VII. Cyclic algebras. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129VII.1. Cyclic algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129VII.2. Central simple algebras over local fields . . . . . . . . . . . . . . . . . . . . . . . . . . . 137VII.3. Central simple algebras over number fields. . . . . . . . . . . . . . . . . . . . . . . . . 139VII.4. Cyclic algebras of prime degree over number fields . . . . . . . . . . . . . . . . . 141VII.5. Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144VII.6. Cyclic algebras and perfect codes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150VII.7. Optimality of some perfect codes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163
Chapter VIII. Central simple algebras of degree 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . 165VIII.1. A theorem of Albert . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165VIII.2. Structure of central simple algebras of degree 4 . . . . . . . . . . . . . . . . . . . 168VIII.3. Albert’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176VIII.4. Codes over biquadratic crossed products . . . . . . . . . . . . . . . . . . . . . . . . . . 178Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187
Chapter IX. Central simple algebras with unitary involutions . . . . . . . . . . . . . . . 189IX.1. Basic concepts. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189IX.2. The corestriction algebra. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191IX.3. Existence of unitary involutions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 198IX.4. Unitary involutions on crossed products. . . . . . . . . . . . . . . . . . . . . . . . . . . 203IX.5. Unitary space-time coding. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 228
Appendix A. Tensor products. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231A.1. Tensor product of vector spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231A.2. Basic properties of the tensor product . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235A.3. Tensor product of k-algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242
Appendix B. A glimpse of number theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 249B.1. Absolute values . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 249B.2. Factorization of ideals in number fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253B.3. Absolute values on number fields and completion . . . . . . . . . . . . . . . . . . . . 262
Appendix C. Complex ideal lattices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 265C.1. Generalities on hermitian lattices. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 265C.2. Complex ideal lattices. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 266
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 271
Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 275
Foreword
Mathematics continually surprises and delights us with how useful its most abstractbranches turn out to be in the real world. Indeed, physicist Eugene Wigner’s mem-orable phrase1 “The unreasonable effectiveness of mathematics” captures a criticalaspect 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 inmulti-antenna wireless communication of the late 1990s would have predicted thatdivision algebras would turn out to be vital in the deployment of multi-antennacommunication. Yet, once performance criteria for space-time codes (as codingschemes for multi-antenna environments are called) were developed and phrased asa problem of design of matrices, it was completely natural that division algebrasshould arise as a solution of the design problem. The fundamental performancecriterion ask for n× n matrices Mi such that the difference of any two of the Mi isof full rank. To anyone who has worked with division algebras, the solution simplyleaps out: any division algebra of index n embeds into the n × n matrices over asuitable field, and the matrices arising from the embedding naturally satisfy thiscriterion.
But there is more. Not only did division algebras turn out to be the most naturalcontext in which to solve the fundamental design problem above, they also proved tobe the correct objects to satisfy various other performance criteria that were devel-oped. For instance, a second, and critical, performance criterion called the codinggain criterion turned out to be naturally satisfied by considering division algebrasover number fields and using natural Z-orders within them that arise from ringsof integers of maximal subfields. Other criteria (for instance “DMG optimality,”“good shaping,” “information-losslessness” to name just a few) all turned out to besatisfied by considering suitable orders inside suitable division algebras over numberfields. Indeed, this exemplifies another phenomenon Wigner describes: after sayingthat “mathematical concepts turn up in entirely unexpected connections,” he goeson to say that “they often permit an unexpectedly close and accurate descriptionof the phenomena in these connections.” The match between division algebras andthe requirement of space-time codes is simply uncanny.
The subject of multi-antenna communication has several unsolved mathematicalproblems still, for instance, in the area of decoding for large numbers of antennas.Nevertheless, division algebras are already being deployed for practical two-antenna
1Eugene P. Wigner, The unreasonable effectiveness of mathematics in the natural sciences,Comm. Pure Appl. Math., 13 Feb. 1960, 1–14
vii
viii FOREWORD
systems, and codes based on them are now part of various standards of the Insti-tute of Electrical and Electronics Engineers (IEEE). It would behoove a studentof mathematics, therefore, to know something about the applicability of divisionalgebras while studying their theory; in parallel, it is vital for a communicationsengineer working in coding for multiple-antenna wireless to know something aboutdivision algebras.
Berhuy and Oggier have written a charming text on division algebras and their ap-plication to multiple-antenna wireless communication. There is a wealth of exam-ples here, particularly over number fields and local fields, with explicit calculations,that one does not see in other texts on the subject. By pairing almost every chapterwith a discussion of issues from wireless communication, the authors have given avery concrete flavor to the subject of division algebras. The book can be studiedprofitably not just by a graduate student in mathematics, but also by a mathe-matically sophisticated coding theorist. I suspect therefore that this book will findwide acceptability in both the mathematics and the space-time coding communityand will help cross-communication between the two. I applaud the authors’ effortsbehind this very enjoyable book.
B.A. Sethuraman
Northridge, California
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Index
k-algebra
center of a, 4central, 8definition, 3morphism, 3
quaternion, 8simple, 7split, 9
absolute discriminant, 259absolute value
p-adic, 262
archimedean, 249definition, 249discrete, 250equivalence, 249
extension, 251non-archimedean, 249
absolutevalue
extensiontotally ramified, 251ramification index, 251ramified, 251
residual degree, 251unramified, 251
bimodule, 82
Brauer equivalence, 46Brauer group, 81
relative, 82
canonical involution, 228centralizer, 31coboundary, 113cocycle, 104
codebook, 13coding gain, 14coherence interval, 12
coherent, 13cohomologous cocycles, 113corestriction, 195
crossed product, 107cyclic algebra, 130
decomposition group, 261
degree, 46
different ideal, 259differential modulation, 209discriminant ideal, 259
diversity, 14
elementary tensor, 233exponent, 95
fading matrix, 11Frobenius map, 253
fully diverse code, 14
Goldman element, 86
Hasse symbol, 138
ideal (ramification)inert, 255
ramification index, 255ramified, 255
tamely ramified, 255totally ramified, 255totally split, 255
unramified, 255wildly ramified, 255
index, 46
information symbol, 12inner automorphism, 43involution
definition of an, 189of the first kind, 189of the second kind, 189
local parameter, 251
MIMO, 11
moduledefinition, 35finitely generated, 36
free, 37morphism, 36rank, 41
non-coherent, 209norm of an ideal
275
276 INDEX
absolute norm, 258relative norm, 258
number field, 253
opposite algebra, 34
place, 249complex, 262finite, 262real, 262
prime idealsresidual degree, 255
ramification groups, 261rate, 14, 17reduced characteristic polynomial, 63reduced norm, 66
reduced trace, 66residue field, 250restriction map, 82ring of integers, 253
Sandwich morphism, 35semilinear map, 191simple
module, 39SNR, 13space-time codes, 13splitting field, 53subalgebra
definition, 3subfield, 9submodule, 36
tensor productof algebras, 5, 243of vector spaces, 231
trace form, 76
valuation ring, 250
Selected Published Titles in This Series
191 Gregory Berhuy and Frederique Oggier, An Introduction to Central Simple Algebrasand Their Applications to Wireless Communication, 2013
187 Nassif Ghoussoub and Amir Moradifam, Functional Inequalities: New Perspectivesand New Applications, 2013
186 Gregory Berkolaiko and Peter Kuchment, Introduction to Quantum Graphs, 2013
185 Patrick Iglesias-Zemmour, Diffeology, 2013
184 Frederick W. Gehring and Kari Hag, The Ubiquitous Quasidisk, 2012
183 Gershon Kresin and Vladimir Maz’ya, Maximum Principles and Sharp Constants forSolutions of Elliptic and Parabolic Systems, 2012
182 Neil A. Watson, Introduction to Heat Potential Theory, 2012
181 Graham J. Leuschke and Roger Wiegand, Cohen-Macaulay Representations, 2012
180 Martin W. Liebeck and Gary M. Seitz, Unipotent and Nilpotent Classes in SimpleAlgebraic Groups and Lie Algebras, 2012
179 Stephen D. Smith, Subgroup complexes, 2011
178 Helmut Brass and Knut Petras, Quadrature Theory, 2011
177 Alexei Myasnikov, Vladimir Shpilrain, and Alexander Ushakov,Non-commutative Cryptography and Complexity of Group-theoretic Problems, 2011
176 Peter E. Kloeden and Martin Rasmussen, Nonautonomous Dynamical Systems, 2011
175 Warwick de Launey and Dane Flannery, Algebraic Design Theory, 2011
174 Lawrence S. Levy and J. Chris Robson, Hereditary Noetherian Prime Rings andIdealizers, 2011
173 Sariel Har-Peled, Geometric Approximation Algorithms, 2011
172 Michael Aschbacher, Richard Lyons, Stephen D. Smith, and Ronald Solomon,The Classification of Finite Simple Groups, 2011
171 Leonid Pastur and Mariya Shcherbina, Eigenvalue Distribution of Large RandomMatrices, 2011
170 Kevin Costello, Renormalization and Effective Field Theory, 2011
169 Robert R. Bruner and J. P. C. Greenlees, Connective Real K-Theory of FiniteGroups, 2010
168 Michiel Hazewinkel, Nadiya Gubareni, and V. V. Kirichenko, Algebras, Ringsand Modules, 2010
167 Michael Gekhtman, Michael Shapiro, and Alek Vainshtein, Cluster Algebras andPoisson Geometry, 2010
166 Kyung Bai Lee and Frank Raymond, Seifert Fiberings, 2010
165 Fuensanta Andreu-Vaillo, Jose M. Mazon, Julio D. Rossi, and J. JulianToledo-Melero, Nonlocal Diffusion Problems, 2010
164 Vladimir I. Bogachev, Differentiable Measures and the Malliavin Calculus, 2010
163 Bennett Chow, Sun-Chin Chu, David Glickenstein, Christine Guenther, James
Isenberg, Tom Ivey, Dan Knopf, Peng Lu, Feng Luo, and Lei Ni, The Ricci Flow:Techniques and Applications: Part III: Geometric-Analytic Aspects, 2010
162 Vladimir Maz′ya and Jurgen Rossmann, Elliptic Equations in Polyhedral Domains,2010
161 Kanishka Perera, Ravi P. Agarwal, and Donal O’Regan, Morse Theoretic Aspectsof p-Laplacian Type Operators, 2010
160 Alexander S. Kechris, Global Aspects of Ergodic Group Actions, 2010
159 Matthew Baker and Robert Rumely, Potential Theory and Dynamics on theBerkovich Projective Line, 2010
For a complete list of titles in this series, visit theAMS Bookstore at www.ams.org/bookstore/survseries/.
SURV/191
Central simple algebras arise naturally in many areas of mathematics. They are closely connected with ring theory, but are also important in representation theory, algebraic geometry and number theory.
Recently, surprising applications of the theory of central simple algebras have arisen in the context of coding for wire-less communication. The exposition in the book takes advantage of this serendipity, presenting an introduction to the theory of central simple algebras intertwined with its applications to coding theory. Many results or constructions from the standard theory are presented in classical form, but with a focus on explicit techniques and examples, often from coding theory.
Topics covered include quaternion algebras, splitting fields, the Skolem-Noether Theorem, the Brauer group, crossed products, cyclic algebras and algebras with a unitary involution. Code constructions give the opportunity for many examples and explicit computations.
This book provides an introduction to the theory of central algebras accessible to graduate students, while also presenting topics in coding theory for wireless commu-nication for a mathematical audience. It is also suitable for coding theorists interested in learning how division algebras may be useful for coding in wireless communication.
www.ams.orgAMS on the Web
For additional information and updates on this book, visit
www.ams.org/bookpages/surv-191
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