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The first edition of this book is a collection of a series of
lectures given by Professor Victor Kac at the TIFR, Mumbai, India
in December 1985 and January 1986. These lectures focus on the idea
of a highest weight representation, which goes through four
different incarnations.The first is the canonical commutation
relations of the infinite dimensional Heisenberg Algebra (=
oscillator algebra). The second is the highest weight
representations of the Lie algebra g of infinite matrices, along
with their applications to the theory of soliton equations,
discovered by Sato and Date, Jimbo, Kashiwara and Miwa. The third
is the unitary highest weight representations of the current (=
affine Kac-Moody) algebras. These Lie algebras appear in the
lectures in connection to the Sugawara construction, which is the
main tool in the study of the fourth incarnation of the main idea,
the theory of the highest weight representations of the Virasoro
algebra. In particular, the book provides a complete proof of the
Kac determinant formula, the key result in representation theory of
the Virasoro algebra.The second edition of this book incorporates,
as its first part, the largely unchanged text of the first edition,
while its second part is the collection of lectures on vertex
algebras, delivered by Professor Kac at the TIFR in January 2003.
The basic idea of these lectures was to demonstrate how the key
notions of the theory of vertex algebras - such as quantum fields,
their normal ordered product and lambda-bracket, energy-momentum
field and conformal weight, untwisted and twisted representations -
simplify and clarify the constructions of the first edition of the
book.This book should be very useful for both mathematicians and
physicists. To mathematicians, it illustrates the interaction of
the key ideas of the representation theory of infinite dimensional
Lie algebras and of the theory of vertex algebras; and to
physicists, these theories are turning into an important component
of such domains of theoretical physics as soliton theory, conformal
field theory, the theory of two-dimensional statistical models, and
string theory.
The first edition of this book is a collection of a series of
lectures given by Professor Victor Kac at the TIFR, Mumbai, India
in December 1985 and January 1986. These lectures focus on the idea
of a highest weight representation, which goes through four
different incarnations.The first is the canonical commutation
relations of the infinite dimensional Heisenberg Algebra (=
oscillator algebra). The second is the highest weight
representations of the Lie algebra g of infinite matrices, along
with their applications to the theory of soliton equations,
discovered by Sato and Date, Jimbo, Kashiwara and Miwa. The third
is the unitary highest weight representations of the current (=
affine Kac-Moody) algebras. These Lie algebras appear in the
lectures in connection to the Sugawara construction, which is the
main tool in the study of the fourth incarnation of the main idea,
the theory of the highest weight representations of the Virasoro
algebra. In particular, the book provides a complete proof of the
Kac determinant formula, the key result in representation theory of
the Virasoro algebra.The second edition of this book incorporates,
as its first part, the largely unchanged text of the first edition,
while its second part is the collection of lectures on vertex
algebras, delivered by Professor Kac at the TIFR in January 2003.
The basic idea of these lectures was to demonstrate how the key
notions of the theory of vertex algebras - such as quantum fields,
their normal ordered product and lambda-bracket, energy-momentum
field and conformal weight, untwisted and twisted representations -
simplify and clarify the constructions of the first edition of the
book.This book should be very useful for both mathematicians and
physicists. To mathematicians, it illustrates the interaction of
the key ideas of the representation theory of infinite dimensional
Lie algebras and of the theory of vertex algebras; and to
physicists, these theories are turning into an important component
of such domains of theoretical physics as soliton theory, conformal
field theory, the theory of two-dimensional statistical models, and
string theory.
The main part of this book describes the first semester of the
existence of a successful and now highly popular program for
elementary school students at the Berkeley Math Circle. The topics
discussed in the book introduce the participants to the basics of
many important areas of modern mathematics, including logic,
symmetry, probability theory, knot theory, cryptography, fractals,
and number theory. Each chapter in the first part of this book
consists of two parts. It starts with generously illustrated sets
of problems and hands-on activities. This part is addressed to
young readers who can try to solve problems on their own or to
discuss them with adults. The second part of each chapter is
addressed to teachers and parents. It includes comments on the
topics of the lesson, relates those topics to discussions in other
chapters, and describes the actual reaction of math circle
participants to the proposed activities. The supplementary problems
that were discussed at workshops of Math Circle at Kansas State
University are given in the second part of the book. The book is
richly illustrated, which makes it attractive to its young
audience. In the interest of fostering a greater awareness and
appreciation of mathematics and its connections to other
disciplines and everyday life, MSRI and the AMS are publishing
books in the Mathematical Circles Library series as a service to
young people, their parents and teachers, and the mathematics
profession. Titles in this series are co-published with the
Mathematical Sciences Research Institute (MSRI).
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