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This book is based on the notes of the authors' seminar on
algebraic and Lie groups held at the Department of Mechanics and
Mathematics of Moscow University in 1967/68. Our guiding idea was
to present in the most economic way the theory of semisimple Lie
groups on the basis of the theory of algebraic groups. Our main
sources were A. Borel's paper [34], C. ChevalIey's seminar [14],
seminar "Sophus Lie" [15] and monographs by C. Chevalley [4], N.
Jacobson [9] and J-P. Serre [16, 17]. In preparing this book we
have completely rearranged these notes and added two new chapters:
"Lie groups" and "Real semisimple Lie groups". Several traditional
topics of Lie algebra theory, however, are left entirely
disregarded, e.g. universal enveloping algebras, characters of
linear representations and (co)homology of Lie algebras. A
distinctive feature of this book is that almost all the material is
presented as a sequence of problems, as it had been in the first
draft of the seminar's notes. We believe that solving these
problems may help the reader to feel the seminar's atmosphere and
master the theory. Nevertheless, all the non-trivial ideas, and
sometimes solutions, are contained in hints given at the end of
each section. The proofs of certain theorems, which we consider
more difficult, are given directly in the main text. The book also
contains exercises, the majority of which are an essential
complement to the main contents.
This book gives an exposition of the fundamentals of the theory of
linear representations of ?nite and compact groups, as well as
elements of the t- ory of linear representations of Lie groups. As
an application we derive the Laplace spherical functions. The book
is based on lectures that I delivered in the framework of the
experimental program at the Mathematics-Mechanics Faculty of Moscow
State University and at the Faculty of Professional Skill
Improvement. My aim has been to give as simple and detailed an
account as possible of the problems considered. The book therefore
makes no claim to completeness. Also, it can in no way give a
representative picture of the modern state of the ?eld under study
as does, for example, the monograph of A. A. Kirillov [3]. For a
more complete acquaintance with the theory of representations of
?nite groups we recommend the book of C. W. Curtis and I. Reiner
[2], and for the theory of representations of Lie groups, that of
M. A. Naimark [6]. Introduction The theory of linear
representations of groups is one of the most widely -
pliedbranchesof algebra. Practically every
timethatgroupsareencountered, their linear representations play an
important role. In the theory of groups itself, linear
representations are an irreplaceable source of examples and a tool
for investigating groups. In the introduction we discuss some
examples and en route we introduce a number of notions of
representation theory. 0. Basic Notions 0. 1.
The aim of the Expositions is to present new and important
developments in pure and applied mathematics. Well established in
the community over more than two decades, the series offers a large
library of mathematical works, including several important
classics. The volumes supply thorough and detailed expositions of
the methods and ideas essential to the topics in question. In
addition, they convey their relationships to other parts of
mathematics. The series is addressed to advanced readers interested
in a thorough study of the subject. Editorial Board Lev Birbrair,
Universidade Federal do Ceara, Fortaleza, Brasil Walter D. Neumann,
Columbia University, New York, USA Markus J. Pflaum, University of
Colorado, Boulder, USA Dierk Schleicher, Jacobs University, Bremen,
Germany Katrin Wendland, University of Freiburg, Germany Honorary
Editor Victor P. Maslov, Russian Academy of Sciences, Moscow,
Russia Titles in planning include Yuri A. Bahturin, Identical
Relations in Lie Algebras (2019) Yakov G. Berkovich, Lev G.
Kazarin, and Emmanuel M. Zhmud', Characters of Finite Groups,
Volume 2 (2019) Jorge Herbert Soares de Lira, Variational Problems
for Hypersurfaces in Riemannian Manifolds (2019) Volker Mayer,
Mariusz Urbanski, and Anna Zdunik, Random and Conformal Dynamical
Systems (2021) Ioannis Diamantis, Bostjan Gabrovsek, Sofia
Lambropoulou, and Maciej Mroczkowski, Knot Theory of Lens Spaces
(2021)
The aim of the Expositions is to present new and important
developments in pure and applied mathematics. Well established in
the community over more than two decades, the series offers a large
library of mathematical works, including several important
classics. The volumes supply thorough and detailed expositions of
the methods and ideas essential to the topics in question. In
addition, they convey their relationships to other parts of
mathematics. The series is addressed to advanced readers interested
in a thorough study of the subject. Editorial Board Lev Birbrair,
Universidade Federal do Ceara, Fortaleza, Brasil Walter D. Neumann,
Columbia University, New York, USA Markus J. Pflaum, University of
Colorado, Boulder, USA Dierk Schleicher, Jacobs University, Bremen,
Germany Katrin Wendland, University of Freiburg, Germany Honorary
Editor Victor P. Maslov, Russian Academy of Sciences, Moscow,
Russia Titles in planning include Yuri A. Bahturin, Identical
Relations in Lie Algebras (2019) Yakov G. Berkovich, Lev G.
Kazarin, and Emmanuel M. Zhmud', Characters of Finite Groups,
Volume 2 (2019) Jorge Herbert Soares de Lira, Variational Problems
for Hypersurfaces in Riemannian Manifolds (2019) Volker Mayer,
Mariusz Urbanski, and Anna Zdunik, Random and Conformal Dynamical
Systems (2021) Ioannis Diamantis, Bostjan Gabrovsek, Sofia
Lambropoulou, and Maciej Mroczkowski, Knot Theory of Lens Spaces
(2021)
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