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Undergraduate Texts in Mathematics Editors S. Axler K.A. Ribet

Undergraduate Texts in Mathematics Abbott: Understanding Analysis. Anglin: Mathematics: A Concise History and Philosophy. Readings in Mathematics. Anglin/Lambek: The Heritage of Thales. Readings in Mathematics. Apostol: Introduction to Analytic Number Theory. Second edition. Armstrong: Basic Topology. Armstrong: Groups and Symmetry. Axler: Linear Algebra Done Right. Second edition. Beardon: Limits: A New Approach to Real Analysis. Bak/Newman: Complex Analysis. Second edition. Banchoff/Wermer: Linear Algebra Through Geometry. Second edition. Beck/Robins: Computing the Continuous Discretely Daepp/Gorkin: Reading, Writing, and Proving: A Closer Look at Mathematics. Devlin: The Joy of Sets: Fundamentals of-Contemporary Set Theory. Second edition. Dixmier: General Topology. Driver: Why Math? Ebbinghaus/Flum/Thomas: Mathematical Logic. Second edition. Edgar: Measure, Topology, and Fractal Geometry. Second edition. Elaydi: An Introduction to Difference Equations. Third edition. Erd˜os/Sur´anyi: Topics in the Theory of Numbers. Estep: Practical Analysis on One Variable. Exner: An Accompaniment to Higher Mathematics. Exner: Inside Calculus. Fine/Rosenberger: The Fundamental Theory of Algebra. Berberian: A First Course in Real Analysis. Bix: Conics and Cubics: A Concrete Introduction to Fischer: Intermediate Real Analysis. Flanigan/Kazdan: Calculus Two: Linear and Algebraic Curves. Second edition. Br`emaud: An Introduction to Probabilistic Nonlinear Functions. Second edition. Fleming: Functions of Several Variables. Second Modeling. Bressoud: Factorization and Primality Testing. Bressoud: Second Year Calculus. Readings in Mathematics. Brickman: Mathematical Introduction to Linear Programming and Game Theory. Browder: Mathematical Analysis: An Introduction. Buchmann: Introduction to Cryptography. Second Edition. Buskes/van Rooij: Topological Spaces: From Distance to Neighborhood. Callahan: The Geometry of Spacetime: An Introduction to Special and General Relavitity. Carter/van Brunt: The Lebesgue– Stieltjes Integral: A Practical Introduction. edition. Foulds: Combinatorial Optimization for Undergraduates. Foulds: Optimization Techniques: An Introduction. Franklin: Methods of Mathematical Economics. Frazier: An Introduction to Wavelets Through Linear Algebra. Gamelin: Complex Analysis. Ghorpade/Limaye: A Course in Calculus and Real Analysis Gordon: Discrete Probability. Hairer/Wanner: Analysis by Its History. Readings in Mathematics. Halmos: Finite-Dimensional Vector Spaces. Cederberg: A Course in Modern Geometries. Second edition. Second edition. Chambert-Loir: A Field Guide to Algebra Childs: A Concrete Introduction to Higher Algebra. Second edition. Halmos: Naive Set Theory. H¨ammerlin/Hoffmann: Numerical Mathematics. Readings in Mathematics. Harris/Hirst/Mossinghoff: Combinatorics and Chung/AitSahlia: Elementary Probability Theory: With Stochastic Processes and an Introduction to Mathematical Finance. Fourth edition. Cox/Little/O’Shea: Ideals, Varieties, and Algorithms. Second edition. Graph Theory. Hartshorne: Geometry: Euclid and Beyond. Hijab: Introduction to Calculus and Classical Analysis. Second edition. Hilton/Holton/Pedersen: Mathematical Croom: Basic Concepts of Algebraic Topology. Cull/Flahive/Robson: Difference Equations. From Reﬂections: In a Room with Many Mirrors. Hilton/Holton/Pedersen: Mathematical Vistas: Rabbits to Chaos From a Room with Many Windows. Curtis: Linear Algebra: An Introductory Approach. Iooss/Joseph: Elementary Stability and Bifurcation Fourth edition. Theory. Second Edition. (continued after index)

John Stillwell Naive Lie Theory 123

John Stillwell Department of Mathematics University of San Francisco San Francisco, CA 94117 USA stillwell@usfca.edu Editorial Board S. Axler Mathematics Department San Francisco State University San Francisco, CA 94132 USA axler@sfsu.edu K.A. Ribet Department of Mathematics University of California at Berkeley Berkeley, CA 94720 USA ribet@math.berkeley.edu ISBN: 978-0-387-78214-0 DOI: 10.1007/978-0-387-78214-0 e-ISBN: 978-0-387-78215-7 Library of Congress Control Number: 2008927921 Mathematics Subject Classiﬁcation (2000): 22Exx:22E60 c 2008 Springer Science+Business Media, LLC All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer Science+Business Media, LLC, 233 Spring Street, New York, NY 10013, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use in this publication of trade names, trademarks, service marks, and similar terms, even if they are not identiﬁed as such, is not to be taken as an expression of opinion as to whether or not they are subject to proprietary rights. Printed on acid-free paper 9 8 7 6 5 4 3 2 1 springer.com

To Paul Halmos In Memoriam

Preface It seems to have been decided that undergraduate mathematics today rests on two foundations: calculus and linear algebra. These may not be the best foundations for, say, number theory or combinatorics, but they serve quite well for undergraduate analysis and several varieties of undergradu- ate algebra and geometry. The really perfect sequel to calculus and linear algebra, however, would be a blend of the two—a subject in which calcu- lus throws light on linear algebra and vice versa. Look no further! This perfect blend of calculus and linear algebra is Lie theory (named to honor the Norwegian mathematician Sophus Lie—pronounced “Lee ”). So why is Lie theory not a standard undergraduate topic? The problem is that, until recently, Lie theory was a subject for mature mathematicians or else a tool for chemists and physicists. There was no Lie theory for novice mathematicians. Only in the last few years have there been serious attempts to write Lie theory books for undergraduates. These books broke through to the undergraduate level by making some sensible compromises with generality; they stick to matrix groups and mainly to the classical ones, such as rotation groups of n-dimensional space. In this book I stick to similar subject matter. The classical groups are introduced via a study of rotations in two, three, and four dimensions, which is also an appropriate place to bring in complex numbers and quater- nions. From there it is only a short step to studying rotations in real, complex, and quaternion spaces of any dimension. In so doing, one has introduced the classical simple Lie groups, in their most geometric form, using only basic linear algebra. Then calculus intervenes to ﬁnd the tan- gent spaces of the classical groups—their Lie algebras—and to move back and forth between the group and its algebra via the log and exponential functions. Again, the basics sufﬁce: single-variable differentiation and the Taylor series for ex and log(1 + x). vii

viii Preface Where my book diverges from the others is at the next level, the mirac- ulous level where one discovers that the (curved) structure of a Lie group is almost completely captured by the structure of its (ﬂat) Lie algebra. At this level, the other books retain many traces of the sophisticated approach to Lie theory. For example, they rely on deep ideas from outside Lie theory, such as the inverse function theorem, existence theorems for ODEs, and representation theory. Even inside Lie theory, they depend on the Killing form and the whole root system machine to prove simplicity of the classical Lie algebras, and they use everything under the sun to prove the Campbell– Baker–Hausdorff theorem that lifts structure from the Lie algebra to the Lie group. But actually, proving simplicity of the classical Lie algebras can be done by basic matrix arithmetic, and there is an amazing elementary proof of Campbell–Baker–Hausdorff due to Eichler [1968]. The existence of these little-known elementary proofs convinced me that a naive approach to Lie theory is possible and desirable. The aim of this book is to carry it out—developing the central concepts and results of Lie theory by the simplest possible methods, mainly from single-variable calculus and linear algebra. Familiarity with elementary group theory is also desirable, but I provide a crash course on the basics of group theory in Sections 2.1 and 2.2. The naive approach to Lie theory is due to von Neumann [1929], and it is now possible to streamline it by using standard results of undergraduate mathematics, particularly the results of linear algebra. Of course, there is a downside to naivet´e. It is probably not powerful enough to prove some of the results for which Lie theory is famous, such as the classiﬁcation of the simple Lie algebras and the discovery of the ﬁve exceptional algebras.1 To compensate for this lack of technical power, the end-of-chapter discussions introduce important results beyond those proved in the book, as part of an informal sketch of Lie theory and its history. It is also true that the naive methods do not afford the same insights as more sophisticated methods. But they offer another insight that is often undervalued—some important theorems are not as difﬁcult as they look! I think that all mathematics students appreciate this kind of insight. In any case, my approach is not entirely naive. A certain amount of topology is essential, even in basic Lie theory, and in Chapter 8 I take 1I say so from painful experience, having entered Lie theory with the aim of under- standing the exceptional groups. My opinion now is that the Lie theory that precedes the classiﬁcation is a book in itself.

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