Algebraic Structures In Integrability Foreword By Victor Kac

Author : Vladimir V Sokolov
Publisher : World Scientific
Page : 346 pages
File Size : 45,5 Mb
Release : 2020-06-05
Category : Science
ISBN : 9789811219665

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Algebraic Structures In Integrability: Foreword By Victor Kac by Vladimir V Sokolov Pdf

Relationships of the theory of integrable systems with various branches of mathematics are extremely deep and diverse. On the other hand, the most fundamental exactly integrable systems often have applications in theoretical physics. Therefore, many mathematicians and physicists are interested in integrable models.The book is intelligible to graduate and PhD students and can serve as an introduction to separate sections of the theory of classical integrable systems for scientists with algebraic inclinations. For the young, the book can serve as a starting point in the study of various aspects of integrability, while professional algebraists will be able to use some examples of algebraic structures, which appear in the theory of integrable systems, for wide-ranging generalizations.The statements are formulated in the simplest possible form. However, some ways of generalization are indicated. In the proofs, only essential points are mentioned, while for technical details, references are provided. The focus is on carefully selected examples. In addition, the book proposes many unsolved problems of various levels of complexity. A deeper understanding of every chapter of the book may require the study of more rigorous and specialized literature.

Author : Vladimir Sokolov
Publisher : Unknown
Page : 400 pages
File Size : 44,9 Mb
Release : 2020-05-26
Category : Science
ISBN : 9811219648

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Algebraic Structures in Integrability by Vladimir Sokolov Pdf

Relationships of the theory of integrable systems with various branches of mathematics are extremely deep and diverse. On the other hand, the most fundamental exactly integrable systems often have applications in theoretical physics. Therefore, many mathematicians and physicists are interested in integrable models. The book is intelligible to graduate and PhD students and can serve as an introduction to separate sections of the theory of classical integrable systems for scientists with algebraic inclinations. For the young, the book can serve as a starting point in the study of various aspects of integrability, while professional algebraists will be able to use some examples of algebraic structures, which appear in the theory of integrable systems, for wide-ranging generalizations. The statements are formulated in the simplest possible form. However, some ways of generalization are indicated. In the proofs, only essential points are mentioned, while for technical details, references are provided. The focus is on carefully selected examples. In addition, the book proposes many unsolved problems of various levels of complexity. A deeper understanding of every chapter of the book may require the study of more rigorous and specialized literature.

Author : Amit K. Roy-Chowdhury
Publisher : CRC Press
Page : 367 pages
File Size : 49,8 Mb
Release : 2021-01-04
Category : Mathematics
ISBN : 9781000116786

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Lie Algebraic Methods in Integrable Systems by Amit K. Roy-Chowdhury Pdf

Over the last thirty years, the subject of nonlinear integrable systems has grown into a full-fledged research topic. In the last decade, Lie algebraic methods have grown in importance to various fields of theoretical research and worked to establish close relations between apparently unrelated systems. The various ideas associated with Lie algebra and Lie groups can be used to form a particularly elegant approach to the properties of nonlinear systems. In this volume, the author exposes the basic techniques of using Lie algebraic concepts to explore the domain of nonlinear integrable systems. His emphasis is not on developing a rigorous mathematical basis, but on using Lie algebraic methods as an effective tool. The book begins by establishing a practical basis in Lie algebra, including discussions of structure Lie, loop, and Virasor groups, quantum tori and Kac-Moody algebras, and gradation. It then offers a detailed discussion of prolongation structure and its representation theory, the orbit approach-for both finite and infinite dimension Lie algebra. The author also presents the modern approach to symmetries of integrable systems, including important new ideas in symmetry analysis, such as gauge transformations, and the "soldering" approach. He then moves to Hamiltonian structure, where he presents the Drinfeld-Sokolov approach, the Lie algebraic approach, Kupershmidt's approach, Hamiltonian reductions and the Gelfand Dikii formula. He concludes his treatment of Lie algebraic methods with a discussion of the classical r-matrix, its use, and its relations to double Lie algebra and the KP equation.

Author : Victor G. Kac
Publisher : Springer Science & Business Media
Page : 267 pages
File Size : 50,5 Mb
Release : 2013-11-09
Category : Mathematics
ISBN : 9781475713824

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Infinite Dimensional Lie Algebras by Victor G. Kac Pdf

Author : Decio Levi,Pavel Winternitz,Ravil I. Yamilov
Publisher : American Mathematical Society, Centre de Recherches Mathématiques
Page : 520 pages
File Size : 55,8 Mb
Release : 2023-01-23
Category : Mathematics
ISBN : 9780821843543

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Continuous Symmetries and Integrability of Discrete Equations by Decio Levi,Pavel Winternitz,Ravil I. Yamilov Pdf

This book on integrable systems and symmetries presents new results on applications of symmetries and integrability techniques to the case of equations defined on the lattice. This relatively new field has many applications, for example, in describing the evolution of crystals and molecular systems defined on lattices, and in finding numerical approximations for differential equations preserving their symmetries. The book contains three chapters and five appendices. The first chapter is an introduction to the general ideas about symmetries, lattices, differential difference and partial difference equations and Lie point symmetries defined on them. Chapter 2 deals with integrable and linearizable systems in two dimensions. The authors start from the prototype of integrable and linearizable partial differential equations, the Korteweg de Vries and the Burgers equations. Then they consider the best known integrable differential difference and partial difference equations. Chapter 3 considers generalized symmetries and conserved densities as integrability criteria. The appendices provide details which may help the readers' understanding of the subjects presented in Chapters 2 and 3. This book is written for PhD students and early researchers, both in theoretical physics and in applied mathematics, who are interested in the study of symmetries and integrability of difference equations.

Author : Victor G. Kac
Publisher : American Mathematical Soc.
Page : 216 pages
File Size : 43,6 Mb
Release : 1998
Category : Mathematical physics
ISBN : UOM:39015041992150

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Vertex Algebras for Beginners by Victor G. Kac Pdf

Based on courses given by the author at MIT and at Rome University in spring 1997, this book presents an introduction to algebraic aspects of conformal field theory. It includes material on the foundations of a rapidly growing area of algebraic conformal theory.

Author : Timothée Marquis
Publisher : Unknown
Page : 331 pages
File Size : 45,9 Mb
Release : 2018
Category : MATHEMATICS
ISBN : 3037196874

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An Introduction to Kac–Moody Groups Over Fields by Timothée Marquis Pdf

The interest for Kac–Moody algebras and groups has grown exponentially in the past decades, both in the mathematical and physics communities, and with it also the need for an introductory textbook on the topic. The aims of this book are twofold: - to offer an accessible, reader-friendly and self-contained introduction to Kac–Moody algebras and groups; - to clean the foundations and to provide a unified treatment of the theory. The book starts with an outline of the classical Lie theory, used to set the scene. Part II provides a self-contained introduction to Kac–Moody algebras. The heart of the book is Part III, which develops an intuitive approach to the construction and fundamental properties of Kac–Moody groups. It is complemented by two appendices, respectively offering introductions to affine group schemes and to the theory of buildings. Many exercises are included, accompanying the readers throughout their journey. The book assumes only a minimal background in linear algebra and basic topology, and is addressed to anyone interested in learning about Kac–Moody algebras and/or groups, from graduate (master) students to specialists.

Author : Erik Koelink,Stefan Kolb,Nicolai Reshetikhin,Bart Vlaar
Publisher : American Mathematical Soc.
Page : 362 pages
File Size : 49,5 Mb
Release : 2022-08-30
Category : Education
ISBN : 9781470465209

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Hypergeometry, Integrability and Lie Theory by Erik Koelink,Stefan Kolb,Nicolai Reshetikhin,Bart Vlaar Pdf

This volume contains the proceedings of the virtual conference on Hypergeometry, Integrability and Lie Theory, held from December 7–11, 2020, which was dedicated to the 50th birthday of Jasper Stokman. The papers represent recent developments in the areas of representation theory, quantum integrable systems and special functions of hypergeometric type.

Author : Vincenzo Ancona,Elisabetta Strickland
Publisher : Springer
Page : 307 pages
File Size : 42,5 Mb
Release : 2014-08-27
Category : Mathematics
ISBN : 9783319052540

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Trends in Contemporary Mathematics by Vincenzo Ancona,Elisabetta Strickland Pdf

The topics faced in this book cover a large spectrum of current trends in mathematics, such as Shimura varieties and the Lang lands program, zonotopal combinatorics, non linear potential theory, variational methods in imaging, Riemann holonomy and algebraic geometry, mathematical problems arising in kinetic theory, Boltzmann systems, Pell's equations in polynomials, deformation theory in non commutative algebras. This work contains a selection of contributions written by international leading mathematicians who were speakers at the "INdAM Day", an initiative born in 2004 to present the most recent developments in contemporary mathematics.

Author : Jean-Luc Brylinski,Ranee Brylinski,Victor Guillemin,Victor Kac
Publisher : Springer Science & Business Media
Page : 629 pages
File Size : 49,7 Mb
Release : 2012-12-06
Category : Mathematics
ISBN : 9781461202615

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Lie Theory and Geometry by Jean-Luc Brylinski,Ranee Brylinski,Victor Guillemin,Victor Kac Pdf

This volume, dedicated to Bertram Kostant on the occasion of his 65th birthday, is a collection of 22 invited papers by leading mathematicians working in Lie theory, geometry, algebra, and mathematical physics. Kostant’s fundamental work in all these areas has provided deep new insights and connections, and has created new fields of research. The papers gathered here present original research articles as well as expository papers, broadly reflecting the range of Kostant’s work.

Author : Anonim
Publisher : Unknown
Page : 1448 pages
File Size : 47,9 Mb
Release : 2003
Category : Mathematics
ISBN : UVA:X006180632

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Mathematical Reviews by Anonim Pdf

Author : Anonim
Publisher : Unknown
Page : 814 pages
File Size : 40,5 Mb
Release : 1988
Category : Global analysis (Mathematics)
ISBN : UCAL:B4342570

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Reviews in Global Analysis, 1980-86 as Printed in Mathematical Reviews by Anonim Pdf

Author : Filippo Callegaro,Giovanna Carnovale,Fabrizio Caselli,Corrado De Concini,Alberto De Sole
Publisher : Springer
Page : 0 pages
File Size : 42,7 Mb
Release : 2018-08-30
Category : Mathematics
ISBN : 331986520X

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Perspectives in Lie Theory by Filippo Callegaro,Giovanna Carnovale,Fabrizio Caselli,Corrado De Concini,Alberto De Sole Pdf

Lie theory is a mathematical framework for encoding the concept of symmetries of a problem, and was the central theme of an INdAM intensive research period at the Centro de Giorgi in Pisa, Italy, in the academic year 2014-2015. This book gathers the key outcomes of this period, addressing topics such as: structure and representation theory of vertex algebras, Lie algebras and superalgebras, as well as hyperplane arrangements with different approaches, ranging from geometry and topology to combinatorics.

Author : Ian Malcolm Musson
Publisher : American Mathematical Soc.
Page : 512 pages
File Size : 44,9 Mb
Release : 2012-04-04
Category : Mathematics
ISBN : 9780821868676

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Lie Superalgebras and Enveloping Algebras by Ian Malcolm Musson Pdf

Lie superalgebras are a natural generalization of Lie algebras, having applications in geometry, number theory, gauge field theory, and string theory. This book develops the theory of Lie superalgebras, their enveloping algebras, and their representations. The book begins with five chapters on the basic properties of Lie superalgebras, including explicit constructions for all the classical simple Lie superalgebras. Borel subalgebras, which are more subtle in this setting, are studied and described. Contragredient Lie superalgebras are introduced, allowing a unified approach to several results, in particular to the existence of an invariant bilinear form on $\mathfrak{g}$. The enveloping algebra of a finite dimensional Lie superalgebra is studied as an extension of the enveloping algebra of the even part of the superalgebra. By developing general methods for studying such extensions, important information on the algebraic structure is obtained, particularly with regard to primitive ideals. Fundamental results, such as the Poincare-Birkhoff-Witt Theorem, are established. Representations of Lie superalgebras provide valuable tools for understanding the algebras themselves, as well as being of primary interest in applications to other fields. Two important classes of representations are the Verma modules and the finite dimensional representations. The fundamental results here include the Jantzen filtration, the Harish-Chandra homomorphism, the Sapovalov determinant, supersymmetric polynomials, and Schur-Weyl duality. Using these tools, the center can be explicitly described in the general linear and orthosymplectic cases. In an effort to make the presentation as self-contained as possible, some background material is included on Lie theory, ring theory, Hopf algebras, and combinatorics.

Author : S. Chmutov,Sergeĭ Vasilʹevich Duzhin,J. Mostovoy
Publisher : Cambridge University Press
Page : 521 pages
File Size : 45,8 Mb
Release : 2012-05-24
Category : Mathematics
ISBN : 9781107020832

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Introduction to Vassiliev Knot Invariants by S. Chmutov,Sergeĭ Vasilʹevich Duzhin,J. Mostovoy Pdf

A detailed exposition of the theory with an emphasis on its combinatorial aspects.