Lie groups, Lie algebras, cohomology, and some applications in physics

474 Pages·1998·3.9 MB·English

by  Josi A. de Azcárraga

This book provides a self-contained introduction to the cohomology theory
of Lie groups and algebras and to some of its applications in physics. No
previous knowledge of the mathematical theory is assumed beyond some notions
of Cartan calculus and differential geometry (which are nevertheless reviewed
in the book in detail). The examples, of current interest, are intended to clarify
certain mathematical aspects and to show their usefulness in physical problems.
The topics treated include the differential geometry of Lie groups, fibre bundles
and connections, characteristic classes, index theorems, monopoles, instantons,
extensions of Lie groups and algebras, some applications in supersymmetry,
Chevalley-Eilenberg approach to Lie algebra cohomology, symplectic coho-
mology, jet-bundle approach to variational principles in mechanics, Wess-
Zumino-Witten terms, infinite Lie algebras, the cohomological descent in
mechanics and in gauge theories and anomalies. A list of references is given
in the bibliographical notes section at the end of each chapter which allows the
reader to complement and go beyond the topics covered in the book.
This book will be of interest to graduate students and researchers in theoretical
physics and applied mathematics.

CAMBRIDGE MONOGRAPHS ON
MATHEMATICAL PHYSICS
General Editors: P. V. Landshoff, D. R. Nelson, D. W. Sciama, S. Weinberg
LIE GROUPS, LIE ALGEBRAS, COHOMOLOGY
AND SOME APPLICATIONS IN PHYSICS
Cambridge Monographs on Mathematical Physics
A. M. Anile Relativistic Fluids and Magneto-Fluids
J. A. de Azcarraga and J. M. Izquierdo Lie groups, Lie Algebras, Cohomology and Some
Applications in Physics
J. Bernstein Kinetic Theory in the Early Universe
G. F. Bertsch and R. A. Broglia Oscillations in Finite Quantum Systems
N. D. Birrell and P. C. W. Davies Quantum Fields in Curved Spacet
D. M. Brink Semiclassical Methods in Nucleus-Nucleus Scattering
J. C. Collins Renormalizationt
P. D. B. Collins An Introduction to Regge Theory and High Energy Physics
M. Creutz Quarks, Gluons and Latticest
F. de Felice and C. J. S. Clarke Relativity on Curved Manifolds
B. DeWitt Supermanifolds, 2nd editiont
P. G. O. Freund Introduction to Supersymmetryt
F G. Friedlander The Wave Equation on a Curved Space-Time
J. Fuchs Affine Lie Algebras and Quantum Groups
J. A. H. Futterman, F. A. Handler and R. A. Matzner Scattering from Black Holes
M. Gockeler and T. SchUcker Differential Geometry, Gauge Theories and Gravityt
C. Gomez, M. Ruiz Altaba and G. Sierra Quantum Groups in Two-dimensional Physics
M. B. Green, J. H. Schwarz and E. Witten Superstring Theory, volume 1: Introductiont
M. B. Green, J. H. Schwarz and E. Witten Superstring Theory, volume 2: Loop Amplitudes,
Anomalies and Phenomenologyt
S. W. Hawking and G. F. R. Ellis The Large-Scale Structure of Space- Time
F. lachello and A. Arima The Interacting Boson Model
F. lachello and P. van Isacker The Interacting Boson-Fermion Model
C. Itzykson and J: M. Drouffe Statistical Field Theory, volume 1: From Brownian Motion to
Renormalization and Lattice Gauge Theoryt
C. Itzykson and J: M. Drouffe Statistical Field Theory, volume 2: Strong Coupling, Monte Carlo
Methods, Conformal Field Theory, and Random Systemst
J. I. Kapusta Finite-Temperature Field Theory
V. E. Korepin, A. G. Izergin and N. M. Boguliubov The Quantum Inverse Scattering Method and
Correlation Functions
D. Kramer, H. Stephani, M. A. H. MacCallum and E. Herlt Exact Solutions of Einstein's Field
Equations
N. H. March Liquid Metals: Concepts and Theory
L. O'Raifeartaigh Group Structure of Gauge Theoriest
A. Ozorio de Almeida Hamiltonian Systems: Chaos and Quantizationt
R. Penrose and W. Rindler Spinors and Space-time, volume 1: Two-Spinor Calculus and
Relativistic Fieldst
R. Penrose and W. Rindler Spinors and Space-time, volume 2 : Spinor and Twistor Methods in
Space-Time Geometry
S. Pokorski Gauge Field Theoriest
V. N. Popov Functional Integrals and Collective Excitationst
R. Rivers Path Integral Methods in Quantum Field Theoryt
R. G. Roberts The Structure of the Proton
W. C. Saslaw Gravitational Physics of Stellar and Galactic Systemst
J. M. Stewart Advanced General Relativity
A. Vilenkin and E. P. S. Shellard Cosmic Strings and other Topological Defects
R. S. Ward and R. O. Wells Jr Twistor Geometry and Field Theoriest
t Issued as a paperback
LIE GROUPS,
LIE ALGEBRAS,
COHOMOLOGY AND
SOME APPLICATIONS
IN PHYSICS
JOSE A. DE AZCARRAGA' AND JOSE M. IZQUIERDOt
* Departamento de Fisica Teorica and
IFIC, Centro Mixto CSIC-Universidad de Valencia
Facultad de Fisica, Valencia University
t Departamento de Fisica Teorica,
Valladolid University
AMBRIDGE
UNIVERSITY PRESS
CAMBRIDGE UNIVERSITY PRESS
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Cambridge University Press
The Edinburgh Building, Cambridge CB2 8RU, UK
Published in the United States of America by Cambridge University Press, New York
www.cambridge.org
Information on this title: www.cambridge.org/9780521465014
© Cambridge University Press 1995
This publication is in copyright. Subject to statutory exception
and to the provisions of relevant collective licensing agreements,
no reproduction of any part may take place without the written
permission of Cambridge University Press.
First published 1995
First paperback edition 1998
A catalogue record for this publication is available from the British Library
Library of Congress Cataloguing in Publication data
Azcarraga, J. A. de, 1941-
Lie groups, Lie algebras, cohomology, and some applications in
physics / Jose A. de Azcarraga and Jose M. Izquierdo.
p. cm. - (Cambridge monographs on mathematical physics)
Includes bibliographical references.
ISBN 0 521 46501 X
1. Lie groups. 2. Lie algebras. 3. Homology theory.
4. Mathematical physics. I. Izquierdo, Jose M. II. Title.
III. Series
QC20.7.L54A93 1995
512'.55-dc2O 94-16809 CIP
ISBN 978-0-521-46501-4 hardback
ISBN 978-0-521-59700-5 paperback
Transferred to digital printing 2008
No mirando a nuestro daho,
corremos a rienda suelta
sin parar;
des' que vemos el engano
y queremos dar la vuelta
no hay lugar.
(Jorge Manrique, 1440-1479)
No foe, no dangerous pass, we heed.
Brook no delay - but onward speed
With loosened rein;
And, when the fatal snare is near,
We strive to check our mad career,
But strive in vain.
(Free 1833 translation by
H. Wadsworth Longfellow)

Contents
Preface xiii
1 Lie groups, fibre bundles and Cartan calculus 1
1.1 Introduction: Lie groups and actions of a Lie group on a manifold 2
1.2 Left- (X') and right- (XR) invariant vector fields on a Lie group G 8
1.3 A summary of fibre bundles 12
1.4 Differential forms and Cartan calculus: a review 31
1.5 De Rham cohomology and Hodge-de Rham theory 43
1.6 The dual aspect of Lie groups: invariant differential forms.
Invariant integration measure on G 54
1.7 The Maurer-Cartan equations and the canonical form on a Lie
group G. Bi-invariant measure 58
1.8 Left-invariance and bi-invariance. Bi-invariant metric tensor field
on the group manifold 62
1.9 Applications and examples for Lie groups 65
1.10 The case of super Lie groups: the supertranslation group as an
example 70
1.11 Appendix A: some homotopy groups 73
1.12 Appendix B: the Poincare polynomials of the compact simple
groups 78
Bibliographical notes for chapter 1 81
2 Connections and characteristic classes 84
2.1 Connections on a principal bundle: an outline 84
2.2 Examples of connections 95
2.3 Equivariant forms on a Lie group 101
2.4 Characteristic classes 104
2.5 Chern classes and Chern characters 111
2.6 Chern-Simons forms of the Chern characters 115
2.7 The magnetic monopole 120
ix