Infinite-Dimensional Aspects of Representation Theory and Applications PDF Download

Author: Stephen Berman
Publisher: American Mathematical Soc.
ISBN: 082183701X
Category : Mathematics
Languages : en
Pages : 168

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Book Description
The University of Virginia (Charlottesville) hosted an international conference on Infinite-dimensional Aspects of Representation Theory and Applications. This volume contains papers resulting from the mini-courses and talks given at the meeting. Beyond the techniques and ideas related to representation theory, the book demonstrates connections to number theory, algebraic geometry, and mathematical physics. The specific topics covered include Hecke algebras, quantum groups, infinite-dimensional Lie algebras, quivers, modular representations, and Gromov-Witten invariants. The book is suitable for graduate students and researchers interested in representation theory.

Author: Augustin Banyaga
Publisher: World Scientific
ISBN: 9814488143
Category : Science
Languages : en
Pages : 174

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Book Description
This book constitutes the proceedings of the 2000 Howard conference on “Infinite Dimensional Lie Groups in Geometry and Representation Theory”. It presents some important recent developments in this area. It opens with a topological characterization of regular groups, treats among other topics the integrability problem of various infinite dimensional Lie algebras, presents substantial contributions to important subjects in modern geometry, and concludes with interesting applications to representation theory. The book should be a new source of inspiration for advanced graduate students and established researchers in the field of geometry and its applications to mathematical physics.

Author: Victor G Kac
Publisher: World Scientific
ISBN: 9814507725
Category : Mathematics
Languages : en
Pages : 159

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Book Description
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 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 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.

Author: Boris Khesin
Publisher: Springer Science & Business Media
ISBN: 3540772634
Category : Mathematics
Languages : en
Pages : 304

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Book Description
This monograph gives an overview of various classes of infinite-dimensional Lie groups and their applications in Hamiltonian mechanics, fluid dynamics, integrable systems, gauge theory, and complex geometry. The text includes many exercises and open questions.

Author: Ashok K Raina
Publisher: World Scientific
ISBN: 981452221X
Category : Mathematics
Languages : en
Pages : 250

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Book Description
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.

Author: Minoru Wakimoto
Publisher: World Scientific
ISBN: 9810241291
Category : Mathematics
Languages : en
Pages : 456

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Book Description
The representation theory of affine Lie algebras has been developed in close connection with various areas of mathematics and mathematical physics in the last two decades. There are three excellent books on it, written by Victor G Kac. This book begins with a survey and review of the material treated in Kac's books. In particular, modular invariance and conformal invariance and are explained in more detail. The book then goes further, dealing with some of the recent topics involving the representation theory of affine Lie algebras. Since these topics are important not only in themselves but also in their application to some areas of mathematics and mathematical physics, the book expounds them with examples and detailed calculations.

Author: Palle Jorgensen
Publisher: World Scientific
ISBN: 9811225796
Category : Mathematics
Languages : en
Pages : 253

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Book Description
The purpose of this book is to make available to beginning graduate students, and to others, some core areas of analysis which serve as prerequisites for new developments in pure and applied areas. We begin with a presentation (Chapters 1 and 2) of a selection of topics from the theory of operators in Hilbert space, algebras of operators, and their corresponding spectral theory. This is a systematic presentation of interrelated topics from infinite-dimensional and non-commutative analysis; again, with view to applications. Chapter 3 covers a study of representations of the canonical commutation relations (CCRs); with emphasis on the requirements of infinite-dimensional calculus of variations, often referred to as Ito and Malliavin calculus, Chapters 4-6. This further connects to key areas in quantum physics.

Author: Raymond C. Fabec
Publisher:
ISBN: 9780816042760
Category : Infinite groups
Languages : en
Pages : 428

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Book Description

Author: Caroline Gruson
Publisher: Springer
ISBN: 3319982710
Category : Mathematics
Languages : en
Pages : 223

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Book Description
This text covers a variety of topics in representation theory and is intended for graduate students and more advanced researchers who are interested in the field. The book begins with classical representation theory of finite groups over complex numbers and ends with results on representation theory of quivers. The text includes in particular infinite-dimensional unitary representations for abelian groups, Heisenberg groups and SL(2), and representation theory of finite-dimensional algebras. The last chapter is devoted to some applications of quivers, including Harish-Chandra modules for SL(2). Ample examples are provided and some are revisited with a different approach when new methods are introduced, leading to deeper results. Exercises are spread throughout each chapter. Prerequisites include an advanced course in linear algebra that covers Jordan normal forms and tensor products as well as basic results on groups and rings.

Author: Vyjayanthi Chari
Publisher: American Mathematical Soc.
ISBN: 0821890379
Category : Mathematics
Languages : en
Pages : 222

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Book Description
This volume contains the proceedings of the International Congress of Mathematicians Satellite Conference on Algebraic and Combinatorial Approaches to Representation Theory, held August 12-16, 2010, at the National Institute of Advanced Studies, Bangalore, India, and the follow-up conference held May 18-20, 2012, at the University of California, USA. It contains original research and survey articles on various topics in the theory of representations of Lie algebras, quantum groups and algebraic groups, including crystal bases, categorification, toroidal algebras and their generalisations, vertex algebras, Hecke algebras, Kazhdan-Lusztig bases, $q$-Schur algebras, and Weyl algebras.