Constructions Of Lie Algebras And Their Modules

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Constructions of Lie Algebras and their Modules

George B. Seligman
Constructions of Lie Algebras and their Modules Book PDF

Author : George B. Seligman
Publisher : Springer
Page : 203 pages
File Size : 54,9 Mb
Release : 2006-11-14
Category : Mathematics
ISBN : 9783540388647

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Constructions of Lie Algebras and their Modules by George B. Seligman Pdf

This book deals with central simple Lie algebras over arbitrary fields of characteristic zero. It aims to give constructions of the algebras and their finite-dimensional modules in terms that are rational with respect to the given ground field. All isotropic algebras with non-reduced relative root systems are treated, along with classical anisotropic algebras. The latter are treated by what seems to be a novel device, namely by studying certain modules for isotropic classical algebras in which they are embedded. In this development, symmetric powers of central simple associative algebras, along with generalized even Clifford algebras of involutorial algebras, play central roles. Considerable attention is given to exceptional algebras. The pace is that of a rather expansive research monograph. The reader who has at hand a standard introductory text on Lie algebras, such as Jacobson or Humphreys, should be in a position to understand the results. More technical matters arise in some of the detailed arguments. The book is intended for researchers and students of algebraic Lie theory, as well as for other researchers who are seeking explicit realizations of algebras or modules. It will probably be more useful as a resource to be dipped into, than as a text to be worked straight through.

Rational Constructions of Modules for Simple Lie Algebras

George B. Seligman
Rational Constructions of Modules for Simple Lie Algebras Book PDF

Author : George B. Seligman
Publisher : American Mathematical Soc.
Page : 185 pages
File Size : 48,7 Mb
Release : 1981
Category : Mathematics
ISBN : 9780821850084

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Rational Constructions of Modules for Simple Lie Algebras by George B. Seligman Pdf

This book is directed to researchers in Lie theory and in the theory of linear algebra, associative or otherwise, and to graduate students who have had some background in one or more of these areas.

Introduction to Lie Algebras

K. Erdmann,Mark J. Wildon
Introduction to Lie Algebras Book PDF

Author : K. Erdmann,Mark J. Wildon
Publisher : Springer Science & Business Media
Page : 251 pages
File Size : 40,9 Mb
Release : 2006-09-28
Category : Mathematics
ISBN : 9781846284908

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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.

An Introduction to Lie Groups and Lie Algebras

Alexander A. Kirillov
An Introduction to Lie Groups and Lie Algebras Book PDF

Author : Alexander A. Kirillov
Publisher : Cambridge University Press
Page : 237 pages
File Size : 51,6 Mb
Release : 2008-07-31
Category : Mathematics
ISBN : 9780521889698

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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

Modular Lie Algebras and their Representations

H. Strade
Modular Lie Algebras and their Representations Book PDF

Author : H. Strade
Publisher : CRC Press
Page : 318 pages
File Size : 44,9 Mb
Release : 2020-08-11
Category : Mathematics
ISBN : 9781000103397

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Modular Lie Algebras and their Representations by H. Strade Pdf

This book presents an introduction to the structure and representation theory of modular Lie algebras over fields of positive characteristic. It introduces the beginner to the theory of modular Lie algebras and is meant to be a reference text for researchers.

Modular Lie Algebras

Geoge B. Seligman
Modular Lie Algebras Book PDF

Author : Geoge B. Seligman
Publisher : Springer Science & Business Media
Page : 175 pages
File Size : 54,7 Mb
Release : 2012-12-06
Category : Mathematics
ISBN : 9783642949852

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Modular Lie Algebras by Geoge B. Seligman Pdf

The study of the structure of Lie algebras over arbitrary fields is now a little more than thirty years old. The first papers, to my know ledge, which undertook this study as an end in itself were those of JACOBSON (" Rational methods in the theory of Lie algebras ") in the Annals, and of LANDHERR ("Uber einfache Liesche Ringe") in the Hamburg Abhandlungen, both in 1935. Over fields of characteristic zero, these thirty years have seen the ideas and results inherited from LIE, KILLING, E. CARTAN and WEYL developed and given new depth, meaning and elegance by many contributors. Much of this work is presented in [47, 64, 128 and 234] of the bibliography. For those who find the rationalization for the study of Lie algebras in their connections with Lie groups, satisfying counterparts to these connections have been found over general non-modular fields, with the substitution of the formal groups of BOCHNER [40] (see also DIEUDONNE [108]), or that of the algebraic linear groups of CHEVALLEY [71], for the usual Lie group. In particular, the relation with algebraic linear groups has stimulated the study of Lie algebras of linear transformations. When one admits to consideration Lie algebras over a base field of positive characteristic (such are the algebras to which the title of this monograph refers), he encounters a new and initially confusing scene.

Lie Algebras, Vertex Operator Algebras and Their Applications

Yi-Zhi Huang,Kailash C. Misra
Lie Algebras Vertex Operator Algebras and Their Applications Book PDF

Author : Yi-Zhi Huang,Kailash C. Misra
Publisher : American Mathematical Soc.
Page : 500 pages
File Size : 40,5 Mb
Release : 2007
Category : Lie algebras
ISBN : 9780821839867

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Lie Algebras, Vertex Operator Algebras and Their Applications by Yi-Zhi Huang,Kailash C. Misra Pdf

The articles in this book are based on talks given at the international conference 'Lie algebras, vertex operator algebras and their applications'. The focus of the papers is mainly on Lie algebras, quantum groups, vertex operator algebras and their applications to number theory, combinatorics and conformal field theory.

Automorphic Forms and Lie Superalgebras

Urmie Ray
Automorphic Forms and Lie Superalgebras Book PDF

Author : Urmie Ray
Publisher : Springer Science & Business Media
Page : 293 pages
File Size : 48,6 Mb
Release : 2007-03-06
Category : Mathematics
ISBN : 9781402050107

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Automorphic Forms and Lie Superalgebras by Urmie Ray Pdf

This book provides the reader with the tools to understand the ongoing classification and construction project of Lie superalgebras. It presents the material in as simple terms as possible. Coverage specifically details Borcherds-Kac-Moody superalgebras. The book examines the link between the above class of Lie superalgebras and automorphic form and explains their construction from lattice vertex algebras. It also includes all necessary background information.

Projective Modules over Lie Algebras of Cartan Type

Daniel Ken Nakano
Projective Modules over Lie Algebras of Cartan Type Book PDF

Author : Daniel Ken Nakano
Publisher : American Mathematical Soc.
Page : 84 pages
File Size : 55,5 Mb
Release : 1992
Category : Mathematics
ISBN : 9780821825303

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Projective Modules over Lie Algebras of Cartan Type by Daniel Ken Nakano Pdf

This monograph focuses on extending theorems for the classical Lie algebras in order to determine the structure and representation theory for Lie algebras of Cartan type. More specifically, Nakano investigates the block theory for the restricted universal enveloping algebras of the Lie algebras of Cartan type. The first section employs techniques developed by Holmes and Nakano to prove a Brauer-Humphreys reciprocity law for graded restricted Lie algebras and also to find the decompositions for the intermediate (Verma) modules used in the reciprocity law. The second section uses this information to investigate the structure of projective modules for the Lie algebras of types W and K. The restricted enveloping algebras for these Lie algebras are shown to have one block. Furthermore, Nakano provides a procedure for computing the Cartan invariants for Lie algebras of types W and K, given knowledge about the decomposition of the generalized Verma modules and about the Jantzen matrix of the classical/reductive zero component. Noteworthy for its readability and the continuity of its theme and purpose, this monograph appeals to graduate students and researchers interested in Lie algebras.

Structure of the Standard Modules for the Affine Lie Algebra A1 Superscript (1)

James Lepowsky,Mirko Primc
Structure of the Standard Modules for the Affine Lie Algebra A1 Superscript 1 Book PDF

Author : James Lepowsky,Mirko Primc
Publisher : American Mathematical Soc.
Page : 84 pages
File Size : 47,7 Mb
Release : 1985
Category : Mathematics
ISBN : 9780821850480

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Structure of the Standard Modules for the Affine Lie Algebra A1 Superscript (1) by James Lepowsky,Mirko Primc Pdf

The affine Kac-Moody algebra $A_1^{(1)}$ has recently served as a source of new ideas in the representation theory of infinite-dimensional affine Lie algebras. In particular, several years ago it was discovered that $A_1^{(1)}$ and then a general class of affine Lie algebras could be constructed using operators related to the vertex operators of the physicists' string model. This book develops the calculus of vertex operators to solve the problem of constructing all the standard $A_1^{(1)}$-modules in the homogeneous realization. Aimed primarily at researchers in and students of Lie theory, the book's detailed and concrete exposition makes it accessible and illuminating even to relative newcomers to the field.

Lie Algebras: Theory and Algorithms

W.A. de Graaf
Lie Algebras Theory and Algorithms Book PDF

Author : W.A. de Graaf
Publisher : Elsevier
Page : 407 pages
File Size : 45,6 Mb
Release : 2000-02-04
Category : Mathematics
ISBN : 9780080535456

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Lie Algebras: Theory and Algorithms by W.A. de Graaf Pdf

The aim of the present work is two-fold. Firstly it aims at a giving an account of many existing algorithms for calculating with finite-dimensional Lie algebras. Secondly, the book provides an introduction into the theory of finite-dimensional Lie algebras. These two subject areas are intimately related. First of all, the algorithmic perspective often invites a different approach to the theoretical material than the one taken in various other monographs (e.g., [42], [48], [77], [86]). Indeed, on various occasions the knowledge of certain algorithms allows us to obtain a straightforward proof of theoretical results (we mention the proof of the Poincaré-Birkhoff-Witt theorem and the proof of Iwasawa's theorem as examples). Also proofs that contain algorithmic constructions are explicitly formulated as algorithms (an example is the isomorphism theorem for semisimple Lie algebras that constructs an isomorphism in case it exists). Secondly, the algorithms can be used to arrive at a better understanding of the theory. Performing the algorithms in concrete examples, calculating with the concepts involved, really brings the theory of life.

Lie Algebras, Part 2

E.A. de Kerf,G.G.A. Bäuerle,A.P.E. ten Kroode
Lie Algebras Part 2 Book PDF

Author : E.A. de Kerf,G.G.A. Bäuerle,A.P.E. ten Kroode
Publisher : Elsevier
Page : 553 pages
File Size : 47,9 Mb
Release : 1997-10-30
Category : Science
ISBN : 0080535461

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Lie Algebras, Part 2 by E.A. de Kerf,G.G.A. Bäuerle,A.P.E. ten Kroode Pdf

This is the long awaited follow-up to Lie Algebras, Part I which covered a major part of the theory of Kac-Moody algebras, stressing primarily their mathematical structure. Part II deals mainly with the representations and applications of Lie Algebras and contains many cross references to Part I. The theoretical part largely deals with the representation theory of Lie algebras with a triangular decomposition, of which Kac-Moody algebras and the Virasoro algebra are prime examples. After setting up the general framework of highest weight representations, the book continues to treat topics as the Casimir operator and the Weyl-Kac character formula, which are specific for Kac-Moody algebras. The applications have a wide range. First, the book contains an exposition on the role of finite-dimensional semisimple Lie algebras and their representations in the standard and grand unified models of elementary particle physics. A second application is in the realm of soliton equations and their infinite-dimensional symmetry groups and algebras. The book concludes with a chapter on conformal field theory and the importance of the Virasoro and Kac-Moody algebras therein.

Forty Years Of Algebraic Groups, Algebraic Geometry, And Representation Theory In China: In Memory Of The Centenary Year Of Xihua Cao's Birth

Jie Du,Jianpan Wang,Lei Lin
Forty Years Of Algebraic Groups Algebraic Geometry And Representation Theory In China In Memory Of The Centenary Year Of Xihua Cao s Birth Book PDF

Author : Jie Du,Jianpan Wang,Lei Lin
Publisher : World Scientific
Page : 490 pages
File Size : 44,7 Mb
Release : 2022-10-21
Category : Mathematics
ISBN : 9789811263507

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Forty Years Of Algebraic Groups, Algebraic Geometry, And Representation Theory In China: In Memory Of The Centenary Year Of Xihua Cao's Birth by Jie Du,Jianpan Wang,Lei Lin Pdf

Professor Xihua Cao (1920-2005) was a leading scholar at East China Normal University (ECNU) and a famous algebraist in China. His contribution to the Chinese academic circle is particularly the formation of a world-renowned 'ECNU School' in algebra, covering research areas include algebraic groups, quantum groups, algebraic geometry, Lie algebra, algebraic number theory, representation theory and other hot fields. In January 2020, in order to commemorate Professor Xihua Cao's centenary birthday, East China Normal University held a three-day academic conference. Scholars at home and abroad gave dedications or delivered lectures in the conference. This volume originates from the memorial conference, collecting the dedications of scholars, reminiscences of family members, and 16 academic articles written based on the lectures in the conference, covering a wide range of research hot topics in algebra. The book shows not only scholars' respect and memory for Professor Xihua Cao, but also the research achievements of Chinese scholars at home and abroad.

Lie Groups, Lie Algebras, and Representations

Brian Hall
Lie Groups Lie Algebras and Representations Book PDF

Author : Brian Hall
Publisher : Springer
Page : 452 pages
File Size : 41,5 Mb
Release : 2015-05-11
Category : Mathematics
ISBN : 9783319134673

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Lie Groups, Lie Algebras, and Representations by Brian Hall Pdf

This textbook treats Lie groups, Lie algebras and their representations in an elementary but fully rigorous fashion requiring minimal prerequisites. In particular, the theory of matrix Lie groups and their Lie algebras is developed using only linear algebra, and more motivation and intuition for proofs is provided than in most classic texts on the subject. In addition to its accessible treatment of the basic theory of Lie groups and Lie algebras, the book is also noteworthy for including: a treatment of the Baker–Campbell–Hausdorff formula and its use in place of the Frobenius theorem to establish deeper results about the relationship between Lie groups and Lie algebras motivation for the machinery of roots, weights and the Weyl group via a concrete and detailed exposition of the representation theory of sl(3;C) an unconventional definition of semisimplicity that allows for a rapid development of the structure theory of semisimple Lie algebras a self-contained construction of the representations of compact groups, independent of Lie-algebraic arguments The second edition of Lie Groups, Lie Algebras, and Representations contains many substantial improvements and additions, among them: an entirely new part devoted to the structure and representation theory of compact Lie groups; a complete derivation of the main properties of root systems; the construction of finite-dimensional representations of semisimple Lie algebras has been elaborated; a treatment of universal enveloping algebras, including a proof of the Poincaré–Birkhoff–Witt theorem and the existence of Verma modules; complete proofs of the Weyl character formula, the Weyl dimension formula and the Kostant multiplicity formula. Review of the first edition: This is an excellent book. It deserves to, and undoubtedly will, become the standard text for early graduate courses in Lie group theory ... an important addition to the textbook literature ... it is highly recommended. — The Mathematical Gazette

Introduction to Vertex Operator Algebras and Their Representations

James Lepowsky,Haisheng Li
Introduction to Vertex Operator Algebras and Their Representations Book PDF

Author : James Lepowsky,Haisheng Li
Publisher : Springer Science & Business Media
Page : 330 pages
File Size : 45,7 Mb
Release : 2012-12-06
Category : Mathematics
ISBN : 9780817681869

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Introduction to Vertex Operator Algebras and Their Representations by James Lepowsky,Haisheng Li Pdf

* Introduces the fundamental theory of vertex operator algebras and its basic techniques and examples. * Begins with a detailed presentation of the theoretical foundations and proceeds to a range of applications. * Includes a number of new, original results and brings fresh perspective to important works of many other researchers in algebra, lie theory, representation theory, string theory, quantum field theory, and other areas of math and physics.