Problems Solutions and for Groups, Lie Groups, Lie Algebras Applications with Willi-Hans Steeb Igor Tanski Yorick Hardy University of Johannesburg, South Africa ,~World Scientific NEW JERSEY· LONDON· SINGAPORE· BEIJING· SHANGHAI· HONG KONG· TAIPEI· CHENNAI 1'lIiJli.l'llI'd Ill' World Scienlific Publishing ell. I'll!. Ltd. 5 Toh Tuck Link, Singapore 596224 USA office: 27 Warren Street, Suite 401-402, Ilackensack, NJ 0760 I UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library. PROBLEMS AND SOLUTIONS FOR GROUPS, LIE GROUPS, LIE ALGEBRAS WITH APPLICATIONS Copyright © 2012 by World Scientific Publishing Co. Pte. Ltd. A/I rigllls reserved. This book, or parts thereof; may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, with.out written permission from th.e Publisher. For photocopying of material in this volume, please pay a copying fee through the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA. In this case permission to photocopy is not required from the publisher. ISBN-13 978-981-4383-90-5 ISBN-IO 981-4383-90-2 Printed in Singapore by World Scientific Printers. .I.II<. pi II 'j H)I'>(' or !.IIiI'> hook IS I. . H'lppl.l' II (·ldl('('l.ioll or prohl('IIIS ill group !.I1('(l1''y, Li(' ).';l'Ollp !.I1('()I''y alI(I 1,11' :Ii".(·11I :11'>. 1"IlI'I.II('l'lllOI'(', ella.pLeI' 4 contains appli('al.iolls or I.III'SI' I.()pi('s. 1':11.('11 I'iIIi.PI.('1' cOllLaills LOO completely solved pI'()I)I(,IIIS. Cliaptel's 1,2 ;1,11<1 :\ giv(' a sllol't bilL comprehensive introduction 1.0 1.11(' I.opin; provi<liug all I.IlI' 1'('I(,VHllt definitions and concepts. Chapters 1,2 ;Lllt! :\ also contaiu two solved programming problems and eight supple- 111('lltal''y problems. Chapter 4 contains 10 solved programming problems alit! LO supplementary problems. Chapter 4 covers mainly applications in lilatliclllatical and theoretical physics as well as quantum mechanics, dif fel'ellLial geometry and relativity. Problems cover beginner, advanced and I'('search topics. The problems are self-contained. Accompanying problem books for this book are: Problems and Solutions in Introductory and Advanced Matrix Calculus by Willi-Hans Steeb World Scientific Publishing, Singapore 2006 ISBN 981 256916 2 http://www.worldscibooks.com/mathematics/6202.html Problems and Solutions in Quantum Computing and Quantum Informa tion, third edition by Willi-Hans Steeb and Yorick Hardy World Scientific, Singapore, 2006 ISBN 981-256-916-2 http://www.worldscibooks.com/physics/6077.html The International School for Scientific Computing (ISSC) provides certifi cate courses for this subject. Please contact the author if you want to do this course or other courses of the ISSC. e-mail addresses of the authors: [email protected] yorickhardy©gmail.com Home page of the authors: http://issc.uj.ac.za v Contents Preface v Notation ix 1 Groups 1 2 Lie Groups 71 3 Lie Algebras 143 4 Applications 211 Bibliography 329 Index 339 Vll IX Notation is cleli lied a:-; ( belollgs Lo (3. bet) ~ does HoL belong to (a set) '/ ' 8 subset T of set 5 ,','(1 T the intersection of the sets 5 and T ,','u T the union of the sets 5 and T f/J empty set F<I set of natural numbers 'lL set of integers Q set of rational numbers IF!( set of real numbers IR+ set of nonnegative real numbers e set of complex numbers IR'" n-dimensional Euclidean space space of column vectors with n real components en n-dimensional complex linear space space of column vectors with n complex components H- Hilbert space z A 3{z real part of the complex number z ~z imaginary part of the complex number z Izl modulus of the complex number z Ix + iyl = (x2 + y2)1/2, x, Y E IR f(5) image of the set 5 under the mapping f fog composition of two mappings (f 0 g)(x) = f(g(x)) G group Z(G) center of the group G 'lLn cyclic group {O, 1, ... ,n - I} under addition modulo n GIN factor group Dn nth dihedral group 5n symmetric group on n letters, permutation group An alternating group on n letters, alternating group L Lie algebra en X column vector in the vector space xT transpose of x (row vector) 0 zero (column) vector norm 11·11 x X· Y == x*y t;caJar prodllct (illller prodllcL) in <C" xxy vector product in ]R3 S2 two sphere A,B,C m x n matrices det(A) determinant of a square matrix A tr(A) trace of a square matrix A rank(A) rank of a matrix A AT transpose of the matrix A A conjugate of the matrix A A* conjugate transpose of matrix A At conjugate transpose of matrix A (notation used in physics) inverse of the square matrix A (if it exists) n x n unit matrix unit operator n x n zero matrix matrix product of an m x n matrix A and an n x p matrix B l/l,fl] := AB - BA commutator of square matrices A and B lA, B1+ := AB + BA anticommutator of square matrices A and B Jl0 B Kronecker product of matrices A and B A EB B Direct sum of matrices A and B 6jk Kronecker delta with 6jk = 1 for j = k and 6jk = 0 for j =I k eigenvalue real parameter E t time variable iI Hamilton operator N Number operator 9 metric tensor field real parameter E 1\ exterior product d exterior derivative The Pauli spin matrices are used extensively in the book. They are given by (~ ~), (~ ~i), (~ ~1). CTx := CTy := CTz := In some cases we will also use CTl, CT2 and CT3 to denote CT x, CT y and CT z Chapter 1 Groups A group G is a set of objects {a,b,c,...} (not necessarily countable) to- gether with a binary operation which associates with any ordered pair of elements a,b in G a third element ab in G (closure). The binary operation (called group multiplication) is subject to the following requirements: 1) There exists an element e in G called the identity element (also called neutral element) such that eg =ge=g for all g ∈G. 2) For every g ∈ G there exists an inverse element g−1 in G such that gg−1 =g−1g =e. 3) Associative law. The identity (ab)c=a(bc) is satisfied for all a,b,c∈G. If ab=ba for all a,b∈G we call the group commutative. If G has a finite number of elements it has finite order n(G), where n(G) is thenumberofelements. Otherwise,Ghasinfiniteorder. Lagrange theorem tells us that the order of a subgroup of a finite group is a divisor of the order of the group. If H is a subset of the group G closed under the group operation of G, and ifH isitselfagroupundertheinducedoperation,thenH isasubgroupofG. Let G be a group and S a subgroup. If for all g ∈G, the right coset Sg :={sg : s∈S} 1