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Bombay lectures on highest weight representations of infinite dimensional lie algebras /
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 can...
| Clasificación: | Libro Electrónico |
|---|---|
| Autores principales: | , , |
| Formato: | Electrónico eBook |
| Idioma: | Inglés |
| Publicado: |
Hackensack, New Jersey :
World Scientific,
[2013]
|
| Edición: | Second edition. |
| Colección: | Advanced series in mathematical physics ;
v. 29. |
| Temas: | |
| Acceso en línea: | Texto completo |
Tabla de Contenidos:
- Lecture 1. 1.1. The Lie algebra [symbol] of complex vector fields on the circle. 1.2. Representations V[symbol] of [symbol]. 1.3. Central extensions of [symbol]: the Virasoro algebra
- Lecture 2. 2.1. Definition of positive-energy representations of Vir. 2.2. Oscillator algebra [symbol]. 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[symbol] and gl[symbol] F. Unitarity of highest weight representations of gl[symbol]. 4.4. Representation of a[symbol] 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[symbol] and a[symbol]
- 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[symbol]. 7.2. Defining equations for [symbol] in F[symbol]. 7.3. Differential equations for [symbol] in [symbol]]. 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 det[symbol](c, h) of the contravariant form. 8.2. The determinant det[symbol](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 ā[symbol]. 9.2. Representations of [symbol] in F[symbol]. 9.3. The invariant bilinear form on [symbol]. The action of [symbol] on [symbol]. 9.4. Reduction from a[symbol] to [symbol] and the unitarity of highest weight representations of [symbol].
- Lecture 10. 10.1. Nonabelian generalization of Virasoro operators: the Sugawara construction. 10.2. The Goddard-Kent-Olive construction
- Lecture 11. 11.1. [symbol] 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 [symbol]. 12.2. A tensor product decomposition of some representations of [symbol]. 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 [symbol] 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.