Nilpotent Orbits In Semisimple Lie Algebra
Nilpotent Orbits In Semisimple Lie Algebra

Author: William.M. McGovern

Publisher: Routledge

Published: 2017-10-19

Total Pages: 206

ISBN-13: 1351428683

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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.

Lie Theory
Lie Theory

Author: Jean-Philippe Anker

Publisher: Springer Science & Business Media

Published: 2012-12-06

Total Pages: 341

ISBN-13: 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.

An Introduction to Lie Groups and Lie Algebras
An Introduction to Lie Groups and Lie Algebras

Author: Alexander A. Kirillov

Publisher: Cambridge University Press

Published: 2008-07-31

Total Pages: 237

ISBN-13: 0521889693

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

The Orbit Method in Representation Theory
The Orbit Method in Representation Theory

Author: Dulfo

Publisher: Springer Science & Business Media

Published: 1990-01-01

Total Pages: 244

ISBN-13: 9780817634742

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Ever since its introduction around 1960 by Kirillov, the orbit method has played a major role in representation theory of Lie groups and Lie algebras. This book contains the proceedings of a conference held from August 29 to September 2, 1988, at the University of Copenhagen, about "the orbit method in representation theory." It contains ten articles, most of which are original research papers, by well-known mathematicians in the field, and it reflects the fact that the orbit method plays an important role in the representation theory of semisimple Lie groups, solvable Lie groups, and even more general Lie groups, and also in the theory of enveloping algebras.

Representations and Nilpotent Orbits of Lie Algebraic Systems
Representations and Nilpotent Orbits of Lie Algebraic Systems

Author: Maria Gorelik

Publisher: Springer Nature

Published: 2019-10-18

Total Pages: 553

ISBN-13: 3030235319

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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.

Introduction to Lie Algebras
Introduction to Lie Algebras

Author: K. Erdmann

Publisher: Springer Science & Business Media

Published: 2006-09-28

Total Pages: 254

ISBN-13: 1846284902

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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.

Unipotent and Nilpotent Classes in Simple Algebraic Groups and Lie Algebras
Unipotent and Nilpotent Classes in Simple Algebraic Groups and Lie Algebras

Author: Martin W. Liebeck

Publisher: American Mathematical Soc.

Published: 2012-01-25

Total Pages: 394

ISBN-13: 0821869205

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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.

Algebraic Quotients. Torus Actions and Cohomology. The Adjoint Representation and the Adjoint Action
Algebraic Quotients. Torus Actions and Cohomology. The Adjoint Representation and the Adjoint Action

Author: A. Bialynicki-Birula

Publisher: Springer Science & Business Media

Published: 2013-03-09

Total Pages: 248

ISBN-13: 3662050714

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This is the second volume of the new subseries "Invariant Theory and Algebraic Transformation Groups". The aim of the survey by A. Bialynicki-Birula is to present the main trends and achievements of research in the theory of quotients by actions of algebraic groups. This theory contains geometric invariant theory with various applications to problems of moduli theory. The contribution by J. Carrell treats the subject of torus actions on algebraic varieties, giving a detailed exposition of many of the cohomological results one obtains from having a torus action with fixed points. Many examples, such as toric varieties and flag varieties, are discussed in detail. W.M. McGovern studies the actions of a semisimple Lie or algebraic group on its Lie algebra via the adjoint action and on itself via conjugation. His contribution focuses primarily on nilpotent orbits that have found the widest application to representation theory in the last thirty-five years.