Classification And Identification Of Lie Algebras

Author : Libor Šnob,Pavel Winternitz
Publisher : American Mathematical Soc.
Page : 306 pages
File Size : 53,6 Mb
Release : 2017-04-05
Category : Electronic
ISBN : 9781470436544

Get Book

Classification and Identification of Lie Algebras by Libor Šnob,Pavel Winternitz Pdf

The purpose of this book is to serve as a tool for researchers and practitioners who apply Lie algebras and Lie groups to solve problems arising in science and engineering. The authors address the problem of expressing a Lie algebra obtained in some arbitrary basis in a more suitable basis in which all essential features of the Lie algebra are directly visible. This includes algorithms accomplishing decomposition into a direct sum, identification of the radical and the Levi decomposition, and the computation of the nilradical and of the Casimir invariants. Examples are given for each algorithm. For low-dimensional Lie algebras this makes it possible to identify the given Lie algebra completely. The authors provide a representative list of all Lie algebras of dimension less or equal to 6 together with their important properties, including their Casimir invariants. The list is ordered in a way to make identification easy, using only basis independent properties of the Lie algebras. They also describe certain classes of nilpotent and solvable Lie algebras of arbitrary finite dimensions for which complete or partial classification exists and discuss in detail their construction and properties. The book is based on material that was previously dispersed in journal articles, many of them written by one or both of the authors together with their collaborators. The reader of this book should be familiar with Lie algebra theory at an introductory level.

Author : K. Erdmann,Mark J. Wildon
Publisher : Springer Science & Business Media
Page : 251 pages
File Size : 55,8 Mb
Release : 2006-09-28
Category : Mathematics
ISBN : 9781846284908

Get Book

Introduction to Lie Algebras by K. Erdmann,Mark J. Wildon Pdf

Lie groups and Lie algebras have become essential to many parts of mathematics and theoretical physics, with Lie algebras a central object of interest in their own right. This book provides an elementary introduction to Lie algebras based on a lecture course given to fourth-year undergraduates. The only prerequisite is some linear algebra and an appendix summarizes the main facts that are needed. The treatment is kept as simple as possible with no attempt at full generality. Numerous worked examples and exercises are provided to test understanding, along with more demanding problems, several of which have solutions. Introduction to Lie Algebras covers the core material required for almost all other work in Lie theory and provides a self-study guide suitable for undergraduate students in their final year and graduate students and researchers in mathematics and theoretical physics.

Author : A.L. Onishchik,E.B. Vinberg
Publisher : Springer Science & Business Media
Page : 264 pages
File Size : 42,8 Mb
Release : 1994-07-12
Category : Mathematics
ISBN : 3540546839

Get Book

Lie Groups and Lie Algebras III by A.L. Onishchik,E.B. Vinberg Pdf

A comprehensive and modern account of the structure and classification of Lie groups and finite-dimensional Lie algebras, by internationally known specialists in the field. This Encyclopaedia volume will be immensely useful to graduate students in differential geometry, algebra and theoretical physics.

Author : Alexander A. Kirillov
Publisher : Cambridge University Press
Page : 237 pages
File Size : 40,5 Mb
Release : 2008-07-31
Category : Mathematics
ISBN : 9780521889698

Get Book

An Introduction to Lie Groups and Lie Algebras by Alexander A. Kirillov Pdf

Contemporary introduction to semisimple Lie algebras; concise and informal, with numerous exercises and examples

Author : Alex Harvey,Engelbert L. Schucking
Publisher : Springer Science & Business Media
Page : 546 pages
File Size : 51,5 Mb
Release : 1999
Category : Philosophy
ISBN : 0387985646

Get Book

On Einstein’s Path by Alex Harvey,Engelbert L. Schucking Pdf

This collection of nearly forty essays in honor of the noted physicist and cosmologist Engelbert Schucking spans the gamut of research in Einsteins theory of general relativity and presents a lively and personal account of current work in the field. Indispensable for physicists involved in research in the field, the book includes important chapters by noted theorists such as A. Ashtekar, P.G. Bergmann, J. Ehlers, E.T. Newman, J.V. Narlikar, R. Penrose, D.W. Sciama, J. Stachel, and W. Rindler.

Author : David J Winter
Publisher : Courier Corporation
Page : 162 pages
File Size : 48,9 Mb
Release : 2013-12-01
Category : Mathematics
ISBN : 9780486783468

Get Book

Abstract Lie Algebras by David J Winter Pdf

Solid but concise, this account of Lie algebra emphasizes the theory's simplicity and offers new approaches to major theorems. Author David J. Winter, a Professor of Mathematics at the University of Michigan, also presents a general, extensive treatment of Cartan and related Lie subalgebras over arbitrary fields. Preliminary material covers modules and nonassociate algebras, followed by a compact, self-contained development of the theory of Lie algebras of characteristic 0. Topics include solvable and nilpotent Lie algebras, Cartan subalgebras, and Levi's radical splitting theorem and the complete reducibility of representations of semisimple Lie algebras. Additional subjects include the isomorphism theorem for semisimple Lie algebras and their irreducible modules, automorphism of Lie algebras, and the conjugacy of Cartan subalgebras and Borel subalgebras. An extensive theory of Cartan and related subalgebras of Lie algebras over arbitrary fields is developed in the final chapter, and an appendix offers background on the Zariski topology.

Author : 09 b82
Publisher : Unknown
Page : 49 pages
File Size : 47,8 Mb
Release : 1976
Category : Electronic
ISBN : OCLC:719108765

Get Book

Classification of all simple graded Lie algebras whose Lie algebra is reductive by 09 b82 Pdf

Author : Hans Samelson
Publisher : Springer Science & Business Media
Page : 172 pages
File Size : 40,9 Mb
Release : 2012-12-06
Category : Mathematics
ISBN : 9781461390145

Get Book

Notes on Lie Algebras by Hans Samelson Pdf

(Cartan sub Lie algebra, roots, Weyl group, Dynkin diagram, . . . ) and the classification, as found by Killing and Cartan (the list of all semisimple Lie algebras consists of (1) the special- linear ones, i. e. all matrices (of any fixed dimension) with trace 0, (2) the orthogonal ones, i. e. all skewsymmetric ma trices (of any fixed dimension), (3) the symplectic ones, i. e. all matrices M (of any fixed even dimension) that satisfy M J = - J MT with a certain non-degenerate skewsymmetric matrix J, and (4) five special Lie algebras G2, F , E , E , E , of dimensions 14,52,78,133,248, the "exceptional Lie 4 6 7 s algebras" , that just somehow appear in the process). There is also a discus sion of the compact form and other real forms of a (complex) semisimple Lie algebra, and a section on automorphisms. The third chapter brings the theory of the finite dimensional representations of a semisimple Lie alge bra, with the highest or extreme weight as central notion. The proof for the existence of representations is an ad hoc version of the present standard proof, but avoids explicit use of the Poincare-Birkhoff-Witt theorem. Complete reducibility is proved, as usual, with J. H. C. Whitehead's proof (the first proof, by H. Weyl, was analytical-topological and used the exis tence of a compact form of the group in question). Then come H.

Author : Georgia Benkart,J. Marshall Osborn
Publisher : American Mathematical Soc.
Page : 356 pages
File Size : 43,9 Mb
Release : 1990-11-07
Category : Mathematics
ISBN : 0821854437

Get Book

Lie Algebras and Related Topics by Georgia Benkart,J. Marshall Osborn Pdf

The 1984 classification of the finite-dimensional restricted simple Lie algebras over an algebraically closed field of characteristic $p>7$ provided the impetus for a Special Year of Lie Algebras, held at the University of Wisconsin, Madison, during 1987-88. Work done during the Special Year and afterward put researchers much closer toward a solution of the long-standing problem of determining the finite-dimensional simple Lie algebras over an algebraically closed field of characteristic $p>7$. This volume contains the proceedings of a conference on Lie algebras and related topics, held in May 1988 to mark the end of the Special Year. The conference featured lectures on Lie algebras of prime characteristic, algebraic groups, combinatorics and representation theory, and Kac-Moody and Virasoro algebras. Many facets of recent research on Lie theory are reflected in the papers presented here, testifying to the richness and diversity of this topic.

Author : A L Onishchik,E. B. Vinberg
Publisher : Springer
Page : 0 pages
File Size : 50,9 Mb
Release : 1994-07-26
Category : Mathematics
ISBN : 3662030667

Get Book

Lie Groups and Lie Algebras III by A L Onishchik,E. B. Vinberg Pdf

A comprehensive and modern account of the structure and classification of Lie groups and finite-dimensional Lie algebras, by internationally known specialists in the field. This Encyclopaedia volume will be immensely useful to graduate students in differential geometry, algebra and theoretical physics.

Author : Torsten Schoeneberg
Publisher : Unknown
Page : 131 pages
File Size : 43,6 Mb
Release : 2014
Category : Electronic
ISBN : OCLC:891368088

Get Book

Semisimple Lie Algebras and Their Classification Over P-adic Fields by Torsten Schoeneberg Pdf

Author : B.P. Komrakov,I.S. Krasil'shchik,G.L. Litvinov,A.B. Sossinsky
Publisher : Springer Science & Business Media
Page : 442 pages
File Size : 50,7 Mb
Release : 2012-12-06
Category : Mathematics
ISBN : 9789401152587

Get Book

Lie Groups and Lie Algebras by B.P. Komrakov,I.S. Krasil'shchik,G.L. Litvinov,A.B. Sossinsky Pdf

This collection contains papers conceptually related to the classical ideas of Sophus Lie (i.e., to Lie groups and Lie algebras). Obviously, it is impos sible to embrace all such topics in a book of reasonable size. The contents of this one reflect the scientific interests of those authors whose activities, to some extent at least, are associated with the International Sophus Lie Center. We have divided the book into five parts in accordance with the basic topics of the papers (although it can be easily seen that some of them may be attributed to several parts simultaneously). The first part (quantum mathematics) combines the papers related to the methods generated by the concepts of quantization and quantum group. The second part is devoted to the theory of hypergroups and Lie hypergroups, which is one of the most important generalizations of the classical concept of locally compact group and of Lie group. A natural harmonic analysis arises on hypergroups, while any abstract transformation of Fourier type is gen erated by some hypergroup (commutative or not). Part III contains papers on the geometry of homogeneous spaces, Lie algebras and Lie superalgebras. Classical problems of the representation theory for Lie groups, as well as for topological groups and semigroups, are discussed in the papers of Part IV. Finally, the last part of the collection relates to applications of the ideas of Sophus Lie to differential equations.

Author : Daniel J. Britten,Frank W. Lemire,R. V. Moody
Publisher : American Mathematical Soc.
Page : 398 pages
File Size : 40,9 Mb
Release : 1986
Category : Mathematics
ISBN : 0821860097

Get Book

Lie Algebras and Related Topics by Daniel J. Britten,Frank W. Lemire,R. V. Moody Pdf

As the Proceedings of the 1984 Canadian Mathematical Society's Summer Seminar, this book focuses on some advances in the theory of semisimple Lie algebras and some direct outgrowths of that theory. The following papers are of particular interest: an important survey article by R. Block and R. Wilson on restricted simple Lie algebras, a survey of universal enveloping algebras of semisimple Lie algebras by W. Borho, a course on Kac-Moody Lie algebras by I. G. Macdonald with an extensive bibliography of this field by Georgia Benkart, and a course on formal groups by M. Hazewinkel. Because of the expository surveys and courses, the book will be especially useful to graduate students in Lie theory, as well as to researchers in the field.

Author : Helmut Strade
Publisher : Walter de Gruyter
Page : 392 pages
File Size : 49,5 Mb
Release : 2004
Category : Mathematics
ISBN : 9783110197013

Get Book

Simple Lie Algebras Over Fields of Positive Characteristic: Classifying the absolute toral rank two case by Helmut Strade Pdf

The problem of classifying the finite-dimensional simple Lie algebras over fields of characteristic p > 0 is a long-standing one. Work on this question during the last 45 years has been directed by the Kostrikin-Shafarevich Conjecture of 1966, which states that over an algebraically closed field of characteristic p > 5 a finite-dimensional restricted simple Lie algebra is classical or of Cartan type. This conjecture was proved for p > 7 by Block and Wilson in 1988. The generalization of the Kostrikin-Shafarevich Conjecture for the general case of not necessarily restricted Lie algebras and p > 7 was announced in 1991 by Strade and Wilson and eventually proved by Strade in 1998. The final Block-Wilson-Strade-Premet Classification Theorem is a landmark result of modern mathematics and can be formulated as follows: Every finite-dimensional simple Lie algebra over an algebraically closed field of characteristic p > 3 is of classical, Cartan, or Melikian type. In the three-volume book, the author is assembling the proof of the Classification Theorem with explanations and references. The goal is a state-of-the-art account on the structure and classification theory of Lie algebras over fields of positive characteristic leading to the forefront of current research in this field. This is the second part of the three-volume book about the classification of the simple Lie algebras over algebraically closed fields of characteristics > 3. The first volume contains the methods, examples, and a first classification result. This second volume presents insight in the structure of tori of Hamiltonian and Melikian algebras. Based on sandwich element methods due to Aleksei. I. Kostrikin and Alexander A. Premet and the investigation of absolute toral rank 2 simple Lie algebras over algebraically closed fields of characteristics > 3 is given.

Author : Anthony Henderson
Publisher : Cambridge University Press
Page : 128 pages
File Size : 53,9 Mb
Release : 2012-08-16
Category : Mathematics
ISBN : 9781139561365

Get Book

Representations of Lie Algebras by Anthony Henderson Pdf

This bold and refreshing approach to Lie algebras assumes only modest prerequisites (linear algebra up to the Jordan canonical form and a basic familiarity with groups and rings), yet it reaches a major result in representation theory: the highest-weight classification of irreducible modules of the general linear Lie algebra. The author's exposition is focused on this goal rather than aiming at the widest generality and emphasis is placed on explicit calculations with bases and matrices. The book begins with a motivating chapter explaining the context and relevance of Lie algebras and their representations and concludes with a guide to further reading. Numerous examples and exercises with full solutions are included. Based on the author's own introductory course on Lie algebras, this book has been thoroughly road-tested by advanced undergraduate and beginning graduate students and it is also suited to individual readers wanting an introduction to this important area of mathematics.