无限维李代数的最高权表示HIGHEST WEIGHT REPRESENTATIONS OF INFINITE DIMENSIONAL LIE ALGEBRA

作者： V. Vac等著

出 版 社： Aspen Publishers

出版时间： 2001-12-1字数：版次： 1页数： 145印刷时间： 1987/12/01开本：印次：纸张： 胶版纸I S B N ： 9971503964包装： 平装内容简介

This book is a collection of a series of lectures given by Prof. V Kac at Tata Institute, India in Dec '85 and Jan '86. 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 (= ocillator algebra). The second is the highest weight representations of the Lie algebra gl¥ 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 algebras appear in the lectures twice, in the reduction theory of soliton equations (KP ® KdV) and in the Sugawara construction as 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.

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 to physicists, this theory is turning into an important component of such domains of theoretical physics as soliton theory, theory of two-dimensional statistical models, and string theory.

目录

Preface

Lecture 1

1.1. The Lie algebra d of complex vector fields on the circle

1.2. Representations Va, b of d

1.3. Central extensions of d: the Virasoro algebra

Lecture 2

2.1. Definition of positive-energy representations of Vir

2.2. Oscillator algebra

2.3. Oscillator representations of Vir

Lecture 3

3.1. Complete reducibifity of the oscillator representations of Vir

3.2. Highest weight representations of Vir

3.3. Verma representations M(c, h) and irreducible highest weight representations V(c, h) of Vir

3.4. More (unitary) oscillator representations of Vir

Lecture 4

4.1. Lie algebras of infinite matrices

4.2. Infinite wedge space F and the Dirac positron theory

4.3. Representation of GL andg in F. Unitarity of highest weight representations ofg.

4.4. Representation of a in F

4.5. Representations of Vir in F

Lecture 5

5.1. Boson-fermion correspondence

5.2. Wedging and contracting operators

5.3. Vertex operators. The first part of the boson-fermion correspondence

5.4. Vertex representations ofg and a

Lecture 6

6.1. Schur polynomials

6.2. The second part of the boson-fermion correspondence

6.3. An application: structure of the Virasoro representations forc= 1

Lecture 7

7.1. Orbit of the vacuum vector under GL

7.2. Defining equations for in F(.0)

7.3. Differential equations for in C[x1,x2..]

7.4. Hirota's bilinear equations

7.5. KPhierarchy

7.6. N-soliton solutions

Lecture 8

8.1. Degenerate representations and the determinant detn(C, h) of the contravariant form

8.2. The determinant detn(C' h) as a polynomial in h

8.3. The Kac determinant formula

8.4. Some consequences of the determinant formula for unitarity and degeneracy

Lecture 9

9.1. Representations of loop algebras in

9.2. Representations of g'n inF(m)

9.3. The invariant bilinear form ong. The action of GLn on gn

9.4. Reduction from a to and unitarity of highest weight representations of n

Lecture 10

10.1. Nonabelian generalization of Virasoro operators: the Sugawara construction

10.2. The Goddard-Kent.Olive construction

Lecture 11

11.1. n and its Weyl group

11.2. The Weyl-Kac character formula and Jacobi-Riemann theta functions

Lecture 12

References