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050 4 QA252.3|b.K33 2013eb
072 7 SCI|x004000|2bisacsh
082 04 520|222
090 QA252.3|b.K33 2013eb
100 1 Kac, Victor G.,|d1943-|0https://id.loc.gov/authorities/
names/n83153884|eauthor.
245 10 Bombay lectures on highest weight representations of
infinite dimensional lie algebras /|cVictor G. Kac, Ashok
K. Raina, Natasha Rozhkovskaya.
250 Second edition.
264 1 Hackensack, New Jersey :|bWorld Scientific,|c[2013]
264 4 |c©2014
300 1 online resource (xii, 237 pages).
336 text|btxt|2rdacontent
337 computer|bc|2rdamedia
338 online resource|bcr|2rdacarrier
340 |gpolychrome|2rdacc
347 text file|2rdaft
490 1 Advanced series in mathematical physics ;|vvol. 29
504 Includes bibliographical references (pages 229-234) and
index.
505 0 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].
505 8 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.
520 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,
Kas.
588 0 Print version record.
590 eBooks on EBSCOhost|bEBSCO eBook Subscription Academic
Collection - North America
650 0 Infinite dimensional Lie algebras.|0https://id.loc.gov/
authorities/subjects/sh91003307
650 0 Quantum field theory.|0https://id.loc.gov/authorities/
subjects/sh85109461
650 7 Infinite dimensional Lie algebras.|2fast|0https://
id.worldcat.org/fast/972423
650 7 Quantum field theory.|2fast|0https://id.worldcat.org/fast/
1085105
655 0 Electronic books.
655 4 Electronic books.
700 1 Raina, A. K.,|0https://id.loc.gov/authorities/names/
nr89011497|eauthor.
700 1 Rozhkovskaya, Natasha,|0https://id.loc.gov/authorities/
names/no2013134712|eauthor.
776 08 |iPrint version:|aKac, Victor G., 1943-|tBombay lectures
on highest weight representations of infinite dimensional
lie algebras.|bSecond edition.|dHackensack, New Jersey :
World Scientific, [2013]|z9789814522182|w(DLC) 2013427978
|w(OCoLC)858312870
830 0 Advanced series in mathematical physics ;|0https://
id.loc.gov/authorities/names/n88508540|vv. 29.
856 40 |uhttps://rider.idm.oclc.org/login?url=http://
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856 42 |3Instructions for reading/downloading this eBook|uhttp://
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