Lie Algebras: Finite and Infinite Dimensional Lie Algebras and Applications in Physics

565 Pages·1997·20.5 MB·English

by  E.A. De Kerf

LIE ALGEBRAS
PART 2
FINITE AND INFINITE DIMENSIONAL LIE ALGEBRAS
AND APPLICATIONS IN PHYSICS
STUDIES IN
MATHEMATICAL PHYSICS
VOLUME 7
EDITORS:
E. van GROESEN
Technical University of Twente, Enschede, The Netherlands
E.M. de JAGER
Emeritus, University of Amsterdam, Amsterdam, The Netherlands
NORTH-HOLLAND
LIE ALGEBRAS
PART 2
FINITE AND INFINITE DIMENSIONAL
LIE ALGEBRAS
AND APPLICATIONS IN PHYSICS
E.A. DE KERF
G.G.A. BAUERLE
Institute for Theoretical Physics, University of Amsterdam
Amsterdam, The Netherlands
A.P.E. TEN KROODE
NORTH-HOLLAND
ELSEVIER SCIENCE B.V.
Sara Burgerhartstraat 25
P.O. Box 211, 1000 AE Amsterdam, The Netherlands
ISBN: 0 444 82836 2
(cid:14)9 1997 Elsevier Science B.V. All rights reserved.
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This book is printed on acid-free paper.
Printed in The Netherlands.
Dedicated to
Riet and my teachers, namely,
Profs. J. de Boer, J. Hilgevoord and the late S.A. Wouthuysen
(E. de K.)
Hennle and Paula
(G.B.)
Madelon
(A. ten K.)
This Page Intentionally Left Blank
Preface from the editors
On the cover of their textbook "Lie Algebras, Finite and Infinite Di-
mensional Lie Algebras and Applications in Physics, part I" the authors
say: "Symmetry makes the world go round! The structure of the laws
in physics is largely based on symmetries". There will be hardly any
mathematical physicist who does not agree with the second part of
this statement. The study of symmetries is part of mathematics and
the study of physical laws, in particular conservation laws, is part of
physics. As stated in the editorial preface of the above-mentioned vol-
ume, the first in our series "Studies in Mathematical Physics", it is the
endeavour of the editors and the publisher to stimulate the publication
of actual developments in mathematical physics, with the emphasis on
ideas and methods that are fundamental, interesting and innovating as
well in mathematics as in physics. The series is intended for graduate
students and researchers who want to get acquainted with new fields
without being forced to discover their way in the literature scattered
over many journals.
"Lie Algebras, part I" covers a major part of the theory of Kac-
Moody algebras, stressing primarily their mathematical structure but
only in the last chapter a few applications in physics are discussed.
However, from the outset the authors planned a second volume in or-
der to treat also representations and more applications. This intention
has resulted in the present textbook which is-in view of the applica-
tions in physics- an essential companion to "Lie Algebras, part I". The
editors are happy with the publication of the present volume since the
two volumes together meet so well the purpose of our series "Studies
in Mathematical Physics". The first volume got a very satisfactory
response from the mathematical physics community and this makes it
more than worthwhile that this second volume is now at our disposal.
To give an impression of the contents of the present book it is de-
sirable to supplement this with a short survey of the contents of "Lie
vii
Vlo.lol
Algebras, part I", because the two books are complementary to each
other; in fact, there are in part II several cross-references to part I.
The first five chapters of part I give the general background material
on Lie algebras. Chapters 6-9 are devoted to the structure of finite-
dimensional complex semisimple Lie algebras, their root systems, root
and Dynkin diagrams and the classification of these algebras. The fol-
lowing chapters 10-14 deal with Kac-Moody algebras involving, among
other topics, the generalized Caftan matrix, Serre's construction, the
Cartan-Killing form, root systems, Dynkin diagrams and the classifica-
tion of Kac-Moody algebras, in particular those of affine type. The last
part of the book has two chapters concerning real and imaginary roots
and the root system of untwisted affine Kac-Moody algebras. The
book concludes with chapter 17, where a preview on some applications
in physics is presented.
The exposition of the theory of Kac-Moody algebras is continued in
the present volume. First of all, an explicit construction of (untwisted)
affine Kac-Moody algebras as extensions of loop algebras by a central
element and a derivation is presented in chapter 19. As a preparation
for this chapter, extensions of Lie algebras are discussed in a general
framework in chapter 18.
Chapters 20-23 deal with the representation theory of Kac-Moody
algebras. In fact, some of the concepts of representation theory are
developed in a somewhat more general context, namely that of alge-
bras with a triangular decomposition, of which Kac-Moody algebras
are special examples. The Virasoro algebra is another example and
this more general setup will be advantageous in chapter 28, where
some aspects of the representation theory of the Virasoro algebra are
discussed.
Chapter 20 introduces the basic notion of the representation theory
of Lie algebras with a triangular decomposition, namely that of a high-
est weight representation. For the case of a Kac-Moody algebra there
is the special notion of integvable highest weight representations, which
are characterized in the class of all highest weight representations by
the fact that the Chevalley generators ei and fi are locally nilpotent
operators. In this case much more can be said about the weight system.
In particular, the notion of weight chains and the action of the Weyl
group on the weight system is discussed. Integrable highest weight
representations are treated in chapter 21.
Chapter 22 introduces the generalized Casimir operator, an impor-
tant tool in the representation theory of Kac-Moody algebras, in par-
ticular for the computation of the dimensions of weight spaces, which
ix
is the main topic of chapter 23. Moreover, it is shown that highest
weight representations for algebras with a triangular decomposition
with a highest weight whose restriction to the real Cartan subalgebra
is real (i.e. A E H~t ) can be naturally equipped with a hermitian form.
In the case of integrable highest weight representations for Kac-Moody
algebras this hermitian form is in fact positive definite, i.e. it is an inner
product and the representation space is a (pre-)Hilbert space. This ex-
plains their importance in physical applications. In fact, the integrable
highest weight representations for a finite-dimensional semisimple Lie
algebra coincide with the finite-dimensional irreducible representations
of these algebras, a fact which is demonstrated in chapter 21.
As an application of the representation theory of finite-dimensional
semisimple Lie algebras, the standard and grand unified models of ele-
mentary particle physics are discussed in chapter 24.
A second application, this time of infinite-dimensional Lie algebras,
is found in the theory of soliton equations in chapter 27. The point of
view adopted there is that soliton equations have infinite-dimensional
symmetry groups, which transform solutions into each other. For the
celebrated KP-hierarchy of soliton equations the relevant group is a
central extension of the infinite matrix group GL(cx~, C). As a prepa-
ration for this the theory of the infinite matrix algebra gl(cr C), its
completion gl(co, C) and its central extension g(A~) ~- gl(oo, C) | Cc
is treated in chapter 25. Among other things, explicit vertex ope-
rator constructions for the fundamental integrable representations of
g(A~) are given there. In chapter 26 it is explained that the affine
Kac-Moody algebras g(A (1)) can be naturally embedded in g(A~).
Apart from explicit constructions of their fundamental representations
in terms of the vertex operators constructed for g(Ar this also leads
to subgroups of the central extension of GL(co, C), which are symme-
try groups of KdV-type hierarchies of soliton equations (see chapter
27).
The last application of infinite-dimensional Lie algebras is in the
realm of 2D conformal field theory. This is the topic of chapter 28,
where it is explained that the conformal algebra of n-dimensional Eu-
clidean space is always finite-dimensional except for n - 2, where it is
infinite-dimensional and consists of two commuting copies of the Witt
algebra. The basic ingredients of 2D conformal field theory turn out
to be unitary highest weight representations for the Virasoro algebra,
the universal central extension of the Witt algebra. With this in mind
some highlights of the representation theory of the Virasoro algebra