Bombay Lectures on Highest Weight Representations of Infinite Dimensional Lie Algebras.
Title:
Bombay Lectures on Highest Weight Representations of Infinite Dimensional Lie Algebras.
Author:
Kac, Victor G.
ISBN:
9789814522205
Personal Author:
Edition:
2nd ed.
Physical Description:
1 online resource (250 pages)
Series:
Advanced Series in Mathematical Physics ; v.29
Advanced Series in Mathematical Physics
Contents:
CONTENTS -- Preface -- Preface to the second edition -- Lecture 1 -- 1.1. The Lie algebra d of complex vector fields on the circle -- 1.2. Representations Vα,β of -- 1.3. Central extensions of : the Virasoro algebra -- Lecture 2 -- 2.1. Definition of positive-energy representations of Vir -- 2.2. Oscillator algebra A -- 2.3. Oscillator representations of Vir -- Lecture 3 -- 3.1. Complete reducibility 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. Representations of GL∞ and gl∞ in F. Unitarity of highest weight representations of gl∞ -- 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 operator representations of gl∞ 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 for c = 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. The KP hierarchy -- 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 a∞ -- 9.2. Representations of gl′n in F(m) -- 9.3. The invariant bilinear form on gln. The action of GLn on gln -- 9.4. Reduction from a∞ to sln and the unitarity of highest weight representations of sln -- Lecture 10 -- 10.1. Nonabelian generalization of Virasoro operators: the Sugawara construction -- 10.2. The Goddard-Kent-Olive construction -- Lecture 11 -- 11.1. sl2 and its Weyl group -- 11.2. The Weyl-Kac character formula and Jacobi-Riemann theta functions -- 11.3. A character identity -- Lecture 12 -- 12.1. Preliminaries on sl2 -- 12.2. A tensor product decomposition of some representations of sl2 -- 12.3. Construction and unitarity of the discrete series representations of Vir -- 12.4. Completion of the proof of the Kac determinant formula -- 12.5. On non-unitarity in the region 0 c < 1, h 0 -- Lecture 13 -- 13.1. Formal distributions -- 13.2. Local pairs of formal distributions -- 13.3. Formal Fourier transform -- 13.4. Lambda-bracket of local formal distributions -- Lecture 14 -- 14.1. Completion of U, restricted representations and quantum fields -- 14.2. Normal ordered product -- Lecture 15 -- 15.1. Non-commutative Wick formula -- 15.2. Virasoro formal distribution for free boson -- 15.3. Virasoro formal distribution for neutral free fermions -- 15.4. Virasoro formal distribution for charged free fermions -- Lecture 16 -- 16.1. Conformal weights -- 16.2. Sugawara construction -- 16.3. Bosonization of charged free fermions -- 16.4. Irreducibility theorem for the charge decomposition -- 16.5. An application: the Jacobi triple product identity -- 16.6. Restricted representations of free fermions -- Lecture 17 -- 17.1. Definition of a vertex algebra -- 17.2. Existence Theorem -- 17.3. Examples of vertex algebras -- 17.4. Uniqueness Theorem and n-th product identity -- 17.5. Some constructions.
17.6. Energy-momentum fields -- 17.7. Poisson like definition of a vertex algebra -- 17.8. Borcherds identity -- Lecture 18 -- 18.1. Definition of a representation of a vertex algebra -- 18.2. Representations of the universal vertex algebras -- 18.3. On representations of simple vertex algebras -- 18.4. On representations of simple affine vertex algebras -- 18.5. The Zhu algebra method -- 18.6. Twisted representations -- References -- Index.
Abstract:
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 gl 8 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.
Local Note:
Electronic reproduction. Ann Arbor, Michigan : ProQuest Ebook Central, 2017. Available via World Wide Web. Access may be limited to ProQuest Ebook Central affiliated libraries.
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