Problems and Solutions for Groups, Lie Groups, Lie Algebras With Applications

342 Pages·2012·22.8 MB·English

by  W.H. Steeb

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'
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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
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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