無限維李代數(shù)的最高權(quán)表示HIGHEST WEIGHT REPRESENTATIONS OF INFINITE DIMENSIONAL LIE ALGEBRA

內(nèi)容概要

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.

書籍目錄

PrefaceLecture 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 algebraLecture 2　2.1. Definition of positive-energy representations of Vir　2.2. Oscillator algebra　2.3. Oscillator representations of VirLecture 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 VirLecture 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 FLecture 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 aLecture 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= 1Lecture 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 solutionsLecture 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 degeneracyLecture 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 nLecture 10  10.1. Nonabelian generalization of Virasoro operators: the Sugawara construction  10.2. The Goddard-Kent.Olive constructionLecture 11  11.1. n and its Weyl group  11.2. The Weyl-Kac character formula and Jacobi-Riemann theta functionsLecture 12References

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