Bombay Lectures on Highest Weight Representations of Infinite Dimensional Lie Algebras

The following sections are included:

• PREFACE

• CONTENTS

The following sections are included:

• The Lie algebra d of complex vector fields on the circle

• Representations V_α,β of d

• Central extensions of d: the Virasoro algebra

The following sections are included:

• Definition of positive-energy representations of Vir

• Oscillator algebra 𝒜

• Oscillator representations of Vir

The following sections are included:

• Complete reducibility of the oscillator representations of Vir

• Highest weight representations of Vir

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

• More (unitary) oscillator representations of Vir

The following sections are included:

• Lie algebras of infinite matrices

• Infinite wedge space F and the Dirac positron theory

• Representations of GL_∞ and gℓ_∞ in F. Unitarity of highest weight representations of gℓ_∞

• Representation of a_∞ in F

• Representations of Vir in F

The following sections are included:

• Boson-fermion correspondence

• Wedging and contracting operators

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

• Vertex representations of gℓ_∞ and a_∞

The following sections are included:

• Schur polynomials

• The second part of the boson-fermion correspondence

• An application: structure of the Virasoro representations for c = 1

The following sections are included:

• Orbit of the vacuum vector under GL_∞

• Defining equations for Ω in F⁽⁰⁾

• Differential equations for Ω in ℂ[x₁, x₂,…]

• Hirota's bilinear equations

• The KP hierarchy

• N-soliton solutions

The following sections are included:

• Degenerate representations and the determinant det_n(c, h) of the contravariant form

• The determinant det_n(c, h) as a polynomial in h

• The Kac determinant formula

• Some consequences of the determinant formula for unitarity and degeneracy

The following sections are included:

• Representations of loop algebras in ā_∞

• Representations of in F^(m)

• The invariant bilinear form on . The action of on

• Reduction from a_∞ to and the unitarity of highest weight representations of

The following sections are included:

• Nonabelian generalization of Virasoro operators: the Sugawara construction

• The Goddard-Kent-Olive construction

The following sections are included:

• and its Weyl group

• The Weyl-Kac character formula and Jacobi-Riemann theta functions

• A character identity

The following sections are included:

• Preliminaries on

• A tensor product decomposition of some representation of

• Construction and unitarity of the discrete series representations of Vir

• Completion of the proof of the Kac determinant formula

• On non-unitarity in the region 0 ≤ c < 1, h ≥ 0

The following sections are included:

• REFERENCES