Nilpotent Lie Algebras

Author : M. Goze,Y. Khakimdjanov
Publisher : Springer Science & Business Media
Page : 350 pages
File Size : 49,5 Mb
Release : 2013-11-27
Category : Mathematics
ISBN : 9789401724326

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Nilpotent Lie Algebras by M. Goze,Y. Khakimdjanov Pdf

This volume is devoted to the theory of nilpotent Lie algebras and their applications. Nilpotent Lie algebras have played an important role over the last years both in the domain of algebra, considering its role in the classification problems of Lie algebras, and in the domain of differential geometry. Among the topics discussed here are the following: cohomology theory of Lie algebras, deformations and contractions, the algebraic variety of the laws of Lie algebras, the variety of nilpotent laws, and characteristically nilpotent Lie algebras in nilmanifolds. Audience: This book is intended for graduate students specialising in algebra, differential geometry and in theoretical physics and for researchers in mathematics and in theoretical physics.

Author : Veronique Fischer,Michael Ruzhansky
Publisher : Birkhäuser
Page : 557 pages
File Size : 40,8 Mb
Release : 2016-03-08
Category : Mathematics
ISBN : 9783319295589

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Quantization on Nilpotent Lie Groups by Veronique Fischer,Michael Ruzhansky Pdf

This book presents a consistent development of the Kohn-Nirenberg type global quantization theory in the setting of graded nilpotent Lie groups in terms of their representations. It contains a detailed exposition of related background topics on homogeneous Lie groups, nilpotent Lie groups, and the analysis of Rockland operators on graded Lie groups together with their associated Sobolev spaces. For the specific example of the Heisenberg group the theory is illustrated in detail. In addition, the book features a brief account of the corresponding quantization theory in the setting of compact Lie groups. The monograph is the winner of the 2014 Ferran Sunyer i Balaguer Prize.

Author : William.M. McGovern
Publisher : Routledge
Page : 201 pages
File Size : 45,5 Mb
Release : 2017-10-19
Category : Mathematics
ISBN : 9781351428699

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Nilpotent Orbits In Semisimple Lie Algebra by William.M. McGovern Pdf

Through the 1990s, a circle of ideas emerged relating three very different kinds of objects associated to a complex semisimple Lie algebra: nilpotent orbits, representations of a Weyl group, and primitive ideals in an enveloping algebra. The principal aim of this book is to collect together the important results concerning the classification and properties of nilpotent orbits, beginning from the common ground of basic structure theory. The techniques used are elementary and in the toolkit of any graduate student interested in the harmonic analysis of representation theory of Lie groups. The book develops the Dynkin-Konstant and Bala-Carter classifications of complex nilpotent orbits, derives the Lusztig-Spaltenstein theory of induction of nilpotent orbits, discusses basic topological questions, and classifies real nilpotent orbits. The classical algebras are emphasized throughout; here the theory can be simplified by using the combinatorics of partitions and tableaux. The authors conclude with a survey of advanced topics related to the above circle of ideas. This book is the product of a two-quarter course taught at the University of Washington.

Author : Martin W. Liebeck,Gary M. Seitz
Publisher : American Mathematical Soc.
Page : 394 pages
File Size : 55,6 Mb
Release : 2012-01-25
Category : Mathematics
ISBN : 9780821869208

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Unipotent and Nilpotent Classes in Simple Algebraic Groups and Lie Algebras by Martin W. Liebeck,Gary M. Seitz Pdf

This book concerns the theory of unipotent elements in simple algebraic groups over algebraically closed or finite fields, and nilpotent elements in the corresponding simple Lie algebras. These topics have been an important area of study for decades, with applications to representation theory, character theory, the subgroup structure of algebraic groups and finite groups, and the classification of the finite simple groups. The main focus is on obtaining full information on class representatives and centralizers of unipotent and nilpotent elements. Although there is a substantial literature on this topic, this book is the first single source where such information is presented completely in all characteristics. In addition, many of the results are new--for example, those concerning centralizers of nilpotent elements in small characteristics. Indeed, the whole approach, while using some ideas from the literature, is novel, and yields many new general and specific facts concerning the structure and embeddings of centralizers.

Author : J. F. Humphreys
Publisher : Oxford University Press, USA
Page : 296 pages
File Size : 45,5 Mb
Release : 1996
Category : Language Arts & Disciplines
ISBN : 0198534590

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A Course in Group Theory by J. F. Humphreys Pdf

Each chapter ends with a summary of the material covered and notes on the history and development of group theory.

Author : Roe W. Goodman
Publisher : Springer
Page : 220 pages
File Size : 46,6 Mb
Release : 2006-11-15
Category : Mathematics
ISBN : 9783540375296

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Nilpotent Lie Groups by Roe W. Goodman Pdf

Author : Maria Gorelik,Vladimir Hinich,Anna Melnikov
Publisher : Springer Nature
Page : 553 pages
File Size : 44,8 Mb
Release : 2019-10-18
Category : Mathematics
ISBN : 9783030235314

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Representations and Nilpotent Orbits of Lie Algebraic Systems by Maria Gorelik,Vladimir Hinich,Anna Melnikov Pdf

This volume, a celebration of Anthony Joseph’s fundamental influence on classical and quantized representation theory, explores a wide array of current topics in Lie theory by experts in the area. The chapters are based on the 2017 sister conferences titled “Algebraic Modes of Representations,” the first of which was held from July 16-18 at the Weizmann Institute of Science and the second from July 19-23 at the University of Haifa. The chapters in this volume cover a range of topics, including: Primitive ideals Invariant theory Geometry of Lie group actions Quantum affine algebras Yangians Categorification Vertex algebras This volume is addressed to mathematicians who specialize in representation theory and Lie theory, and who wish to learn more about this fascinating subject.

Author : Roger William Carter
Publisher : Cambridge University Press
Page : 662 pages
File Size : 47,6 Mb
Release : 2005-10-27
Category : Mathematics
ISBN : 0521851386

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Lie Algebras of Finite and Affine Type by Roger William Carter Pdf

This book provides a thorough but relaxed mathematical treatment of Lie algebras.

Author : William.M. McGovern
Publisher : Routledge
Page : 166 pages
File Size : 42,7 Mb
Release : 2017-10-19
Category : Mathematics
ISBN : 9781351428682

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Nilpotent Orbits In Semisimple Lie Algebra by William.M. McGovern Pdf

Through the 1990s, a circle of ideas emerged relating three very different kinds of objects associated to a complex semisimple Lie algebra: nilpotent orbits, representations of a Weyl group, and primitive ideals in an enveloping algebra. The principal aim of this book is to collect together the important results concerning the classification and properties of nilpotent orbits, beginning from the common ground of basic structure theory. The techniques used are elementary and in the toolkit of any graduate student interested in the harmonic analysis of representation theory of Lie groups. The book develops the Dynkin-Konstant and Bala-Carter classifications of complex nilpotent orbits, derives the Lusztig-Spaltenstein theory of induction of nilpotent orbits, discusses basic topological questions, and classifies real nilpotent orbits. The classical algebras are emphasized throughout; here the theory can be simplified by using the combinatorics of partitions and tableaux. The authors conclude with a survey of advanced topics related to the above circle of ideas. This book is the product of a two-quarter course taught at the University of Washington.

Author : Anthony E. Clement,Stephen Majewicz,Marcos Zyman
Publisher : Birkhäuser
Page : 307 pages
File Size : 46,6 Mb
Release : 2017-11-18
Category : Mathematics
ISBN : 9783319662138

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The Theory of Nilpotent Groups by Anthony E. Clement,Stephen Majewicz,Marcos Zyman Pdf

This monograph presents both classical and recent results in the theory of nilpotent groups and provides a self-contained, comprehensive reference on the topic. While the theorems and proofs included can be found throughout the existing literature, this is the first book to collect them in a single volume. Details omitted from the original sources, along with additional computations and explanations, have been added to foster a stronger understanding of the theory of nilpotent groups and the techniques commonly used to study them. Topics discussed include collection processes, normal forms and embeddings, isolators, extraction of roots, P-localization, dimension subgroups and Lie algebras, decision problems, and nilpotent groups of automorphisms. Requiring only a strong undergraduate or beginning graduate background in algebra, graduate students and researchers in mathematics will find The Theory of Nilpotent Groups to be a valuable resource.

Author : Alexander A. Kirillov
Publisher : Cambridge University Press
Page : 237 pages
File Size : 44,8 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

Author : Libor Šnob,Pavel Winternitz
Publisher : American Mathematical Soc.
Page : 306 pages
File Size : 50,8 Mb
Release : 2017-04-05
Category : Electronic
ISBN : 9781470436544

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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 : J.E. Humphreys
Publisher : Springer Science & Business Media
Page : 189 pages
File Size : 54,8 Mb
Release : 2012-12-06
Category : Mathematics
ISBN : 9781461263982

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Introduction to Lie Algebras and Representation Theory by J.E. Humphreys Pdf

This book is designed to introduce the reader to the theory of semisimple Lie algebras over an algebraically closed field of characteristic 0, with emphasis on representations. A good knowledge of linear algebra (including eigenvalues, bilinear forms, euclidean spaces, and tensor products of vector spaces) is presupposed, as well as some acquaintance with the methods of abstract algebra. The first four chapters might well be read by a bright undergraduate; however, the remaining three chapters are admittedly a little more demanding. Besides being useful in many parts of mathematics and physics, the theory of semisimple Lie algebras is inherently attractive, combining as it does a certain amount of depth and a satisfying degree of completeness in its basic results. Since Jacobson's book appeared a decade ago, improvements have been made even in the classical parts of the theory. I have tried to incor porate some of them here and to provide easier access to the subject for non-specialists. For the specialist, the following features should be noted: (I) The Jordan-Chevalley decomposition of linear transformations is emphasized, with "toral" subalgebras replacing the more traditional Cartan subalgebras in the semisimple case. (2) The conjugacy theorem for Cartan subalgebras is proved (following D. J. Winter and G. D. Mostow) by elementary Lie algebra methods, avoiding the use of algebraic geometry.

Author : Joachim Hilgert,Karl-Hermann Neeb
Publisher : Springer Science & Business Media
Page : 742 pages
File Size : 49,9 Mb
Release : 2011-11-06
Category : Mathematics
ISBN : 9780387847948

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Structure and Geometry of Lie Groups by Joachim Hilgert,Karl-Hermann Neeb Pdf

This self-contained text is an excellent introduction to Lie groups and their actions on manifolds. The authors start with an elementary discussion of matrix groups, followed by chapters devoted to the basic structure and representation theory of finite dimensinal Lie algebras. They then turn to global issues, demonstrating the key issue of the interplay between differential geometry and Lie theory. Special emphasis is placed on homogeneous spaces and invariant geometric structures. The last section of the book is dedicated to the structure theory of Lie groups. Particularly, they focus on maximal compact subgroups, dense subgroups, complex structures, and linearity. This text is accessible to a broad range of mathematicians and graduate students; it will be useful both as a graduate textbook and as a research reference.

Author : Jean-Pierre Serre
Publisher : Springer
Page : 180 pages
File Size : 47,6 Mb
Release : 2009-02-07
Category : Mathematics
ISBN : 9783540706342

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Lie Algebras and Lie Groups by Jean-Pierre Serre Pdf

The main general theorems on Lie Algebras are covered, roughly the content of Bourbaki's Chapter I.I have added some results on free Lie algebras, which are useful, both for Lie's theory itself (Campbell-Hausdorff formula) and for applications to pro-Jrgroups. of time prevented me from including the more precise theory of Lack semisimple Lie algebras (roots, weights, etc.); but, at least, I have given, as a last Chapter, the typical case ofal, . This part has been written with the help of F. Raggi and J. Tate. I want to thank them, and also Sue Golan, who did the typing for both parts. Jean-Pierre Serre Harvard, Fall 1964 Chapter I. Lie Algebras: Definition and Examples Let Ie be a commutativering with unit element, and let A be a k-module, then A is said to be a Ie-algebra if there is given a k-bilinear map A x A~ A (i.e., a k-homomorphism A0" A -+ A). As usual we may define left, right and two-sided ideals and therefore quo tients. Definition 1. A Lie algebra over Ie isan algebrawith the following properties: 1). The map A0i A -+ A admits a factorization A ®i A -+ A2A -+ A i.e., ifwe denote the imageof(x, y) under this map by [x, y) then the condition becomes for all x e k. [x, x)=0 2). (lx, II], z]+ny, z), x) + ([z, xl, til = 0 (Jacobi's identity) The condition 1) implies [x,1/]=-[1/, x).