Bombay Lectures on Highest Weight Representations of Infinite Dimensional Lie Algebras

Author: Victor G Kac
Publisher: World Scientific
ISBN: 981452221X
Format: PDF, Kindle

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∞ 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. Contents:Definition of Positive-Energy Representations of VirComplete Reducibility of the Oscillator Representations of VirLie Algebras of Infinite MatricesBoson–Fermion CorrespondenceSchur PolynomialsN-Soliton SolutionsThe Kac Determinant FormulaNonabelian Generalization of Virasoro Operators: The Sugawara ConstructionThe Weyl–Kac Character Formula and Jacobi–Riemann Theta FunctionsCompletion of the Proof of the Kac Determinant FormulaLambda–Bracket of Local Formal DistributionsCompletion of U, Restricted Representations and Quantum FieldsNon-Commutative Wick FormulaConformal WeightsDefinition of a Vertex AlgebraDefinition of a Representation of a Vertex Algebraand other lectures Readership: Mathematicians studying representation theory and theoretical physicists. Keywords:Highest Weight Representations;Virasoro Algebra;Heisenberg Algebra;Infinite-Dimensional Lie Algebras;BosonâFermion Correspondence;Sugawara Construction;Kac Determinant Formula;Vertex Operators;The KP Hierarchy;N-Solitons;Hirota's Bilinear Equations;Vertex Algebras;Quantum Fields;Energy-Momentum Field;Lambda-Bracket;Normal Ordered Product;Conformal Weight;Twisted Representations;Zhu Algebra;Charged Free Fermions;Neutral Free Fermions;Borcherds Identity;Twisted RepresentationsKey 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

Bombay Lectures on Highest Weight Representations of Infinite Dimensional Lie Algebras

Author: Victor G. Kac
Publisher: World Scientific
ISBN: 9789971503963
Format: PDF, ePub, Mobi

This book is a collection of a series of lectures given by Prof. V Kac at Tata Institute, India in Dec '85 and Jan '86. 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 glì 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 algebras appear in the lectures twice, in the reduction theory of soliton equations (KP ? KdV) and in the Sugawara construction as 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.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 to physicists, this theory is turning into an important component of such domains of theoretical physics as soliton theory, theory of two-dimensional statistical models, and string theory.

Bombay Lectures on Highest Weight Representations of Infinite Dimensional Lie Algebras

Author:
Publisher:
ISBN: 9814507725
Format: PDF, Docs

Kac Moody Lie Algebras and Related Topics

Author: Neelacanta Sthanumoorthy
Publisher: American Mathematical Soc.
ISBN: 0821833375
Format: PDF, Docs

This volume is the proceedings of the Ramanujan International Symposium on Kac-Moody Lie algebras and their applications. The symposium provided researchers in mathematics and physics with the opportunity to discuss new developments in this rapidly-growing area of research. The book contains several excellent articles with new and significant results. It is suitable for graduate students and researchers working in Kac-Moody Lie algebras, their applications, and related areas of research.

Vertex Algebras for Beginners

Author: Victor G. Kac
Publisher: American Mathematical Soc.
ISBN: 082181396X
Format: PDF, Mobi

This is a revised and expanded edition of Kac's original introduction to algebraic aspects of conformal field theory, which was published by the AMS in 1996. The volume serves as an introduction to algebraic aspects of conformal field theory, which in the past 15 years revealed a variety of unusual mathematical notions. Vertex algebra theory provides an effective tool to study them in a unified way. In the book, a mathematician encounters new algebraic structures that originated from Einstein's special relativity postulate and Heisenberg's uncertainty principle. A physicist will find familiar notions presented in a more rigorous and systematic way, possibly leading to a better understanding of foundations of quantum physics. This revised edition is based on courses given by the author at MIT and at Rome University in spring 1997. New material is added, including the foundations of a rapidly growing area of algebraic conformal theory. Also, in some places the exposition has been significantly simplified.

W symmetry

Author: P. Bouwknegt
Publisher: World Scientific
ISBN: 9789810217624
Format: PDF, ePub, Mobi

W-symmetry is an extension of conformal symmetry in two dimensions. Since its introduction in 1985, W-symmetry has become one of the central notions in the study of two-dimensional conformal field theory. The mathematical structures that underlie W-symmetry are so-called W-algebras, which are higher-spin extensions of the Virasoro algebra. This book contains a collection of papers on W-symmetry, covering the period from 1985 through 1993. Its main focus is the construction of W-algebras and their representation theory. A recurrent theme is the intimate connection between W-algebras and affine Lie algebras. Some of the applications, in particular W-gravity, are also covered.The significance of this reprint volume is that there are no textbooks entirely devoted to the subject.

Infinite Dimensional Lie Algebras and Groups

Author: V G Kac
Publisher: World Scientific
ISBN: 9814663174
Format: PDF, ePub, Docs

Contents:Integrable Representation of Kac-Moody Algebras: Results and Open Problems (V Chari & A Pressley)Existence of Certain Components in the Tensor Product of Two Integrable Highest Weight Modules for Kac-Moody Algebras (SKumar)Frobenius Action on the B-Cohomology (O Mathieu)Certain Rank Two Subsystems of Kac-Moody Root Systems (J Morita)Lie Groups Associated to Kac-Moody Lie Algebras: An Analytic Approach (E Rodriguez-Carrington)Almost Split-K-Forms of Kac-Moody Algebras (G Rousseau)Global Representations of the Diffeomorphism Groups of the Circle (F Bien)Path Space Realization of the Basic Representation of An(1) (E Date et al)Boson-Fermion Correspondence Over (C De Concini et al)Classification of Modular Invariant Representations of Affine Algebras (V G Kac & M Wakimoto)Standard Monomial Theory for SL2 (V Lakshmibai & C S Seshadri)Some Results on Modular Invariant Representations (S Lu)Current Algebras in 3+1 Space-Time Dimensions (J Mickelson)Standard Representations of An(1) (M Primc)Representations of the Algebra Uq(sI(2)), q-Orthogonal Polynomials and Invariants of Links (A N Kirillov & N Yu Reshetikhin)Infinite Super Grassmannians and Super Plücker Equations (M J Bergvelt)Drinfeld-Sokolov Hierarchies and t-Functions (H J Imbens)Super Boson-Fermion Correspondence of Type B (V G Kac & J W van de Leur)Prym Varieties and Soliton Equations (T Shiota)Polynomial Solutions of the BKP Hierarchy and Projective Representations of Symmetric Groups (Y You)Toward Generalized Macdonald's Identities (D Bernard)Conformal Theories with Non-Linearly Extended Virasoro Symmetries and Lie Algebra Classification (A Bilal & J-LGervais)Extended Conformal Algebras from Kac-Moody Algebras (P Bouwknegt)Meromorphic Conformal Field Theory (P Goddard)Local Extensions of the U(1) Current Algebra and Their Positive Energy Representations (R R Paunov & I T Todorov)Conformal Field Theory on Moduli Family of Stable Curves with Gauge Symmetries (A Tsuchiya & Y Yamada) Readership: Mathematicians and mathematical physicists

An Introduction to the Mathematical Structure of Quantum Mechanics

Author: F. Strocchi
Publisher: World Scientific
ISBN: 9812835229
Format: PDF, Docs

Arising out of the need for Quantum Mechanics (QM) to be part of the common education of mathematics students, this book formulates the mathematical structure of QM in terms of the C*-algebra of observables, which is argued on the basis of the operational definition of measurements and the duality between states and observables.

Gauge Mechanics

Author: L. Mangiarotti
Publisher: World Scientific
ISBN: 9789810236038
Format: PDF, ePub

This book presents in a unified way modern geometric methods in analytical mechanics based on the application of fibre bundles, jet manifold formalism and the related concept of connection. Non-relativistic mechanics is seen as a particular field theory over a one-dimensional base. In fact, the concept of connection is the major link throughout the book. In the gauge scheme of mechanics, connections appear as reference frames, dynamic equations, and in Lagrangian and Hamiltonian formalisms. Non-inertial forces, energy conservation laws and other phenomena related to reference frames are analyzed; that leads us to observable physics. The gauge formulation of classical mechanics is extended to quantum mechanics under different reference frames. Special topics on geometric BRST mechanics, relativistic mechanics and others, together with many examples, are also dealt with.

Topics in Contemporary Mathematical Physics

Author: Kai S Lam
Publisher: World Scientific Publishing Company
ISBN: 981466782X
Format: PDF, ePub, Mobi

This new (second) edition contains a general treatment of quantum field theory (QFT) in a simple scalar field setting in addition to the modern material on the applications of differential geometry and topology, group theory, and the theory of linear operators to physics found in the first edition. All these are introduced without assuming more background on the part of the reader than a good foundation in undergraduate (junior) level mathematical physics. The new material entirely focuses on an introduction to quantum field theory, emphasizing the Feynman path (functional integral) approach to QFT and the renormalization group. With respect to the latter, the focus is on an introduction of its application to critical phenomena in statistical physics, following the outgrowth of the Callan–Symanzik equation originally developed in the context of high energy physics, and the seminal contributions of Kenneth Wilson. One of the overriding aims of the new material is also to draw students' attention to the deep connections between high energy physics and statistical mechanics. The unavoidable technical aspects are explained with a minimum of prerequisite material and jargon, and conceptual understanding is always given prominence before mastery of technical details, but the importance of the latter is never underestimated. Derivational details and motivational discussions are provided in abundance in order to ensure continuity of reading, and to avoid trying the readers' patience.