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作者 Kac, Victor G
書名 Bombay Lectures on Highest Weight Representations of Infinite Dimensional Lie Algebras
出版項 Singapore : World Scientific Publishing Co Pte Ltd, 2013
©2013
國際標準書號 9789814522205 (electronic bk.)
9789814522182
book jacket
版本 2nd ed
說明 1 online resource (250 pages)
text txt rdacontent
computer c rdamedia
online resource cr rdacarrier
系列 Series On Chinese Economics Research ; v.29
Series On Chinese Economics Research
附註 Intro -- 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
Key Features:The first part of the lectures demonstrates four related constructions of highest weight representations of infinite-dimensional algebras: Heisenberg algebra, Lie algebra gl_\infty, affine Kac-Moody algebras and the Virasoro algebra. The constructions originate from theoretical physics and are explained in full detailThe complete proof of the Kac determinant formula is providedThe second part of the lectures demonstrates how the notions of the theory of vertex algebras clarify and simplify the constructions of the first partThe introductory exposition is self-containedMany examples providedCan be used for graduate courses
Description based on publisher supplied metadata and other sources
Electronic reproduction. Ann Arbor, Michigan : ProQuest Ebook Central, 2020. Available via World Wide Web. Access may be limited to ProQuest Ebook Central affiliated libraries
鏈接 Print version: Kac, Victor G. Bombay Lectures on Highest Weight Representations of Infinite Dimensional Lie Algebras Singapore : World Scientific Publishing Co Pte Ltd,c2013 9789814522182
主題 Lie algebras.;Quantum theory
Electronic books
Alt Author Raina, Ashok K
Rozhkovskaya, Natasha
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