Classification and Structure Theory of Lie Algebras of Smooth Sections

Classification and Structure Theory of Lie Algebras of Smooth Sections
Title Classification and Structure Theory of Lie Algebras of Smooth Sections PDF eBook
Author Hasan Gündoğan
Publisher Logos Verlag Berlin GmbH
Pages 172
Release 2011
Genre Mathematics
ISBN 383253024X

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Lie groups and their "derived objects", Lie algebras, appear in various fields of mathematics and physics. At least since the beginning of the 20th century, and after the famous works of Wilhelm Killing, Elie Cartan, Eugenio Elia Levi, Anatoly Malcev and Igor Ado on the structure of finite-dimensional Lie algebras, the classification and structure theory of infinite-dimensional Lie algebras has become an interesting and fairly vast field of interest. This dissertation focusses on the structure of Lie algebras of smooth and k-times differentiable sections of finite-dimensional Lie algebra bundles, which are generalizations of the famous and well-understood affine Kac-Moody algebras. Besides answering the immediate structural questions (center, commutator algebra, derivations, centroid, automorphism group), this work approaches a classification of section algebras by homotopy theory. Furthermore, we determine a universal invariant symmetric bilinear form on Lie algebras of smooth sections and use this form to define a natural central extension which is universal, at least in the case of Lie algebra bundles with compact base manifold.

An Introduction to Lie Groups and Lie Algebras

An Introduction to Lie Groups and Lie Algebras
Title An Introduction to Lie Groups and Lie Algebras PDF eBook
Author Alexander A. Kirillov
Publisher Cambridge University Press
Pages 237
Release 2008-07-31
Genre Mathematics
ISBN 0521889693

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This book is an introduction to semisimple Lie algebras. It is concise and informal, with numerous exercises and examples.

Lectures on Lie Groups

Lectures on Lie Groups
Title Lectures on Lie Groups PDF eBook
Author J. F. Adams
Publisher University of Chicago Press
Pages 192
Release 1982
Genre Mathematics
ISBN 0226005305

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"[Lectures in Lie Groups] fulfills its aim admirably and should be a useful reference for any mathematician who would like to learn the basic results for compact Lie groups. . . . The book is a well written basic text [and Adams] has done a service to the mathematical community."—Irving Kaplansky

Applications of Lie Groups to Differential Equations

Applications of Lie Groups to Differential Equations
Title Applications of Lie Groups to Differential Equations PDF eBook
Author Peter J. Olver
Publisher Springer Science & Business Media
Pages 524
Release 2012-12-06
Genre Mathematics
ISBN 1468402749

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This book is devoted to explaining a wide range of applications of con tinuous symmetry groups to physically important systems of differential equations. Emphasis is placed on significant applications of group-theoretic methods, organized so that the applied reader can readily learn the basic computational techniques required for genuine physical problems. The first chapter collects together (but does not prove) those aspects of Lie group theory which are of importance to differential equations. Applications covered in the body of the book include calculation of symmetry groups of differential equations, integration of ordinary differential equations, including special techniques for Euler-Lagrange equations or Hamiltonian systems, differential invariants and construction of equations with pre scribed symmetry groups, group-invariant solutions of partial differential equations, dimensional analysis, and the connections between conservation laws and symmetry groups. Generalizations of the basic symmetry group concept, and applications to conservation laws, integrability conditions, completely integrable systems and soliton equations, and bi-Hamiltonian systems are covered in detail. The exposition is reasonably self-contained, and supplemented by numerous examples of direct physical importance, chosen from classical mechanics, fluid mechanics, elasticity and other applied areas.

A Guide to Quantum Groups

A Guide to Quantum Groups
Title A Guide to Quantum Groups PDF eBook
Author Vyjayanthi Chari
Publisher Cambridge University Press
Pages 672
Release 1995-07-27
Genre Mathematics
ISBN 9780521558846

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Since they first arose in the 1970s and early 1980s, quantum groups have proved to be of great interest to mathematicians and theoretical physicists. The theory of quantum groups is now well established as a fascinating chapter of representation theory, and has thrown new light on many different topics, notably low-dimensional topology and conformal field theory. The goal of this book is to give a comprehensive view of quantum groups and their applications. The authors build on a self-contained account of the foundations of the subject and go on to treat the more advanced aspects concisely and with detailed references to the literature. Thus this book can serve both as an introduction for the newcomer, and as a guide for the more experienced reader. All who have an interest in the subject will welcome this unique treatment of quantum groups.

Lie Theory

Lie Theory
Title Lie Theory PDF eBook
Author Jean-Philippe Anker
Publisher Springer Science & Business Media
Pages 341
Release 2012-12-06
Genre Mathematics
ISBN 0817681922

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* First of three independent, self-contained volumes under the general title, "Lie Theory," featuring original results and survey work from renowned mathematicians. * Contains J. C. Jantzen's "Nilpotent Orbits in Representation Theory," and K.-H. Neeb's "Infinite Dimensional Groups and their Representations." * Comprehensive treatments of the relevant geometry of orbits in Lie algebras, or their duals, and the correspondence to representations. * Should benefit graduate students and researchers in mathematics and mathematical physics.

Galois' Theory Of Algebraic Equations (Second Edition)

Galois' Theory Of Algebraic Equations (Second Edition)
Title Galois' Theory Of Algebraic Equations (Second Edition) PDF eBook
Author Jean-pierre Tignol
Publisher World Scientific Publishing Company
Pages 325
Release 2015-12-28
Genre Mathematics
ISBN 9814704717

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The book gives a detailed account of the development of the theory of algebraic equations, from its origins in ancient times to its completion by Galois in the nineteenth century. The appropriate parts of works by Cardano, Lagrange, Vandermonde, Gauss, Abel, and Galois are reviewed and placed in their historical perspective, with the aim of conveying to the reader a sense of the way in which the theory of algebraic equations has evolved and has led to such basic mathematical notions as 'group' and 'field'. A brief discussion of the fundamental theorems of modern Galois theory and complete proofs of the quoted results are provided, and the material is organized in such a way that the more technical details can be skipped by readers who are interested primarily in a broad survey of the theory.In this second edition, the exposition has been improved throughout and the chapter on Galois has been entirely rewritten to better reflect Galois' highly innovative contributions. The text now follows more closely Galois' memoir, resorting as sparsely as possible to anachronistic modern notions such as field extensions. The emerging picture is a surprisingly elementary approach to the solvability of equations by radicals, and yet is unexpectedly close to some of the most recent methods of Galois theory.