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Introduction to Lie Groups and Lie Algebras INTRODUCTION TO LIE GROUPS AND LIE ALGEBRAS ARTHUR A. SAGLE RALPH E. WALDE ACADEMIC PRESS New York and London 1973 COPYRIQHT 0 1973, BY ACADEMICPRESS, INC. ALL RIGHTS RESERVED. NO PART OF THIS PUBLICATION MAY BE REPRODUCED OR TRANSMITTED IN ANY FORM OR BY ANY MEANS, ELECTRONIC OR MECHANICAL, INCLUDINQ PHOTOCOPY, RECORDINQ, OR ANY INFORMATION STORAQE AND RETRlEVAL SYSTEM, WITHOUT PERMISSION IN WRITING FROM THE PUBLISHER. ACADEMIC PRESS, INC. 111 Fifth Avenue, New York, New York 10003 United Kingdom Edition published by ACADEMIC PRESS, INC. (LONDON) LTD. 24/28 Oval Road. London NWl LIBRARY OF CONQRRSS CATALWCARD NUMBER: 72 - 77350 AMS (MOS) 1970 Subject Classifications: 22E15, 22E60, 17B05 PRINTED IN THE UNITED STATES OF AMERICA Preface This text is intended for the beginning graduate student with minimal prep- aration. However since Lie groups abstract the analytic properties of matrix groups, the student is expected to have some knowledge of senior level algebra, topology, and analysis as given in some of the references. In Chapter 1 we review some advanced calculus and extend these results to manifolds in Chapter 2. Consequently the reader knowing these results can skip these chapters but should pay attention to the examples on matrix groups. After this the reader probably should follow the order given in the contents noting that the first part of the text is about Lie groups while the algebraic study of Lie algebras begins in Chapter 9. We have not attempted to prove all basic results so the serious student should take the indicated detours to such texts as those by Freudenthal and de Vries, Helgason, Jacobson, or Wolf. In particular the student must develop his own taste in this subject and ours is only one point of view. We are deeply indebted to the many authors and teachers in the subject and to them we express our gratitude. We would like to thank Dr. J. R. Schumi for his assistance, Professors Harry Allen, Charles Conatser, and Earl Taft for their timely suggestions, and the National Science Foundation for their support of mathematics. Also thanks are due to Barb Ketter and Els Sagle for their excellent manuscript preparation and to the people at T.S.V. and at the University of Minnesota for their encouragement. ix CHAPTER 1 SOME CALCULUS We shall present many familiar concepts of differential calculus in the terminology of linear algebra. Thus, for functions from one Euclidean space to another, derivatives are given as linear transformations, higher order derivatives are given as multilinear forms, and Taylor’s series is presented in this terminology. Instead of giving detailed proofs we present many examples involving matrix groups which will be abstracted in later chapters. 1. Basic Notation We now informally review some basic concepts with which the reader should be familiar. Thus let Vdenote an n-dimensional vector space over a field K and let X,, . , X, be a basis of V. Then any point or vector p (or P) in Vcan be uniquely represented by p = cpixi i= 1 and we call the pi E K the coordinates of p relative to the basis X:, . , X, of I/. In particular when we let e, = (0, . , 1,O. .O) with 1 in the kth position and use the basis el, . , , en, then we frequently write p = (PI, . , p,) as a vector in V. For a fixed basis X,, . , X, of V, the functions ui: V+K:p+pi 1 2 1. SOME CALCULUS for i = 1, , . , , n are called the coordinate functions for V relative to the basis X,,. , X, . Thus we obtain a “coordinate system” by a choice of basis in V and in particular obtain the usual coordinate system by choosing the el ,. , en basis of V. Let Wbe an rn-dimensional vector space over K and with Vas above let Hom,(V, W) or just Hom(V, W) denote the set of linear transformations of V into W. Thus T: V+W:X-+T(X) is in Hom(V, W) if T(aX + bY) = aT(X) + bT(Y) for all a, b E K and X, Y E V. In particular Hom(V, W) is a vector space over K of dimension m n relative to the usual definitions: For S, T EHom(V, W)and a, b E K define (US + bT)(X)= aS(X) + bT(X) for all X E V. We shall also use the notation L(V, W) for Hom(V, W) and End(V) for Hom(V, V). Now let K = R, the real numbers, and let V = R” which we regard as the set of all n-tuples X = (x,, . , xn) with xi E R and with the operations ax + bY =(ax, + by1,. *, ax, + by,) for Y = (y,, . , y,,) and a, b E R. With this representation of V we have a natural inner product B:Vx V+R given by the formula n B(X, Y)= c xiyi. i=1 Thus for X,Y, Z E Vand a, b E R,B satisfies (1) B(UX + bY, 2) = aB(X,Z) + bB(Y, 2); (2) w,Y) = B(Y, XI; (3) B(X, X)2 0 and B(X, X)= 0 if and only if X = 0. Any function B : V x V + R satisfying (1) and (2) above is called a symmetric bilinear form and if it also satisfies (3), B is called a positive definite symmetric bilinear form. 1. BASIC NOTATION 3 A norm on a vector space V is a function n : V + R satisfying (I) n(X) 2 0 and n(X) = 0 if and only if X = 0; (2) if a E R and X E I/, then n(aX) = laln(X); (3) n(X + Y) In(X) i- n(Y) for all X, Y E V. We shall also use the notation n(X) = 1x1 = IlXll. In particular, if B is a positive definite symmetric bilinear form on the vector space V over R,then llXll = B(X, X)'/2 is a norm on Vand we have the inequality IB(X9 Y>l 5 I1XIlIlY11. Using a norm on V we can define a metric d on V by 4x9 Y)= IIX - YII for X, Y E V. Thus d satisfies (1) d(X, Y)2 0 and d(X, Y) = 0 if and only if X = Y; (2) 4x9 Y) = d(Y, X); (3) d(X, Y)I d(X, Z) + d(Z, Y>. In particular, with I(X(I= B(X, X)'" =(~x~~)'/~we obtain d(X, Y)= [CCxi - ~i)~]~/~. We now consider some of the topological properties of V = R" which arise from a metric d obtained from a given fixed norm. Thus we define the open ball of radius r with center p by B(p, r) = {X E v : d(p, X) < r} and say subset S of Vis open in Vif for every p E S there exists r > 0 so that the open ball B(p, r) is contained in S. Using this definition we obtain the basic results on the metric topology of R" with which we assume the reader is familiar. Notice that it really does not matter which norm we start with when considering the topological properties of V = R" relative to a metric induced by a norm. Thus if n, and n2 are norms on V, we can show that there exist constants a and b in R so that for all X E V an1(X) 5 nz(X) I bn,(X); that is, n, and n2 are equivalent norms. Thus, if dk is the metric determined by the norm nk,then using the above inequality it is easy to see that the open sets 4 1. SOME CALCULUS of Vrelative to d, are the same as the open sets relative to d2 [DieudonnC, 1960; Lang, 19681. Most often we shall determine a norm n on V by choosing a basis X,,. , X,of Vand for X = CxiXiset n(X) = (cxi2)'". In particular we obtain the usual inner product and norm by taking the basis e,, .. , en of V = R" and call the vector space R" with the topology induced by this norm Euclidean n-space. With the topology in V = R" induced by a norm n as above we now note that V is complete. Thus a sequence of vectors {xk} in V is called a Cauchy sequence if given any E > 0 there exists N so that for all p, q 2 N we have n(xp- xq) < E. We have the result that every Cauchy sequence in V has a limit; that is, V is complete [DieudonnC, 1960; Lang, 19681. Let {xk}be a sequence in Vand let X,, . , X, be a basis of V. Then we can write xk = xk1 xi + *'* + xknxn and note {xk} converges if and only if each sequence {xki}, i = 1, . , n, converges. Thus, by a skillful choice of a basis, it might be obvious that the sequences {xki} converge so that {xk} can be shown to converge easily. Let {xk} be a sequence in V = R". Then the series of vectors cxk in V converges if the sequence {s,,} given by the partial sums P sp = c xk k= 1 converges. Now associated with any series cxk in Vis the seriesof real numbers formed by taking the series of norms of each term. We have the following expected results [DieudonnC, 1960; Lang, 19681. (I) If the series cn(xk)converges in R, then the series cxk converges in the Euclidean space R"; in this case we say cxk converges absolutely. (2) If cxk converges absolutely to the limit a, then the series obtained by any rearrangement of the terms also converges absolutely to a.
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