# Lie Groups, Lie Algebras, and Representations: v. 222 : An Elementary Introduction

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Lie groups, Lie algebras, and representation theory are the main focus of this text. In order to keep the prerequisites to a minimum, the author restricts attention to matrix Lie groups and Lie algebras. This approach keeps the discussion concrete, allows the reader to get to the heart of the subject quickly, and covers all of the most interesting examples. The book also introduces the often-intimidating machinery of roots and the Weyl group in a gradual way, using examples and representation theory as motivation. The text is divided into two parts. The first covers Lie groups and Lie algebras and the relationship between them, along with basic representation theory. The second part covers the theory of semisimple Lie groups and Lie algebras, beginning with a detailed analysis of the representations of SU(3). The author illustrates the general theory with numerous images pertaining to Lie algebras of rank two and rank three, including images of root systems, lattices of dominant integral weights, and weight diagrams. This book is sure to become a standard textbook for graduate students in mathematics and physics with little or no prior exposure to Lie theory.

Brian Hall is an Associate Professor of Mathematics at the University of Notre Dame.

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Brian Hall is an Associate Professor of Mathematics at the University of Notre Dame.

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## Product details

- Hardback | 354 pages
- 157.48 x 236.22 x 22.86mm | 498.95g
- 01 Dec 2004
- Springer-Verlag New York Inc.
- New York, NY, United States
- English
- 1st ed. 2003. Corr. 2nd printing 2004
- biography
- 0387401229
- 9780387401225
- 656,888

## Review quote

From the reviews: "This is an excellent book. It deserves to, and undoubtedly will, become the standard text for early graduate courses in Lie group theory ... . It is clearly written ... . A reader of this book will be rewarded with an excellent understanding of Lie groups ... . Hall's book appears to be genuinely unique in both the organization of the material and the care in which it is presented. This is an important addition to the textbook literature ... . It is highly recommended." (Mark Hunacek, The Mathematical Gazette, March, 2005) "The book is written in a systematic and clear way, each chapter ends with a set of exercises. The book could be valuable for students of mathematics and physics as well as for teachers, for the preparation of courses. It is a nice addition to the existing literature." (EMS-European Mathematical Society Newsletter, September, 2004) "This book differs from most of the texts on Lie Groups in one significant aspect. ... it develops the whole theory on matrix Lie groups. This approach ... will be appreciated by those who find differential geometry difficult to understand. ... each of the eight chapters plus appendix A contain a good collection of exercises. ... I believe that the book fills the gap between the numerous popular books on Lie groups ... is a valuable addition to the collection of any mathematician or physicist interested in the subject." (P.K. Smrz, The Australian Mathematical Society Gazette, Vol. 31 (2), 2004) "This book addresses Lie groups, Lie algebras, and representation theory. ... the author restricts attention to matrix Lie groups and Lie algebras. This approach keeps the discussion concrete, allows the reader to get to the heart of the subject quickly, and covers all the most interesting examples. ... This book is sure to become a standard textbook for graduate students in mathematics and physics with little or no prior exposure to Lie theory." (L'Enseignement Mathematique, Vol. 49 (3-4), 2003) "Though there exist already several excellent text books providing the mathematical basis for all this, introductions aimed at graduate students both in mathematics and physics seem to be rare. So the guiding principle in the planning of the book by Brian Hall ... was to minimize the amount of prerequisites. ... students will benefit from the way the material is presented in this Introduction; for it is elementary and not intimidating, at the same time very systematic, rigorous and modern ... ." (G. Roepstorff, Zentralblatt MATH, Vol. 1026, 2004) "This book is a great find for those who want to learn about Lie groups or Lie algebras and basics of their representation theory. It is a well-written text which introduces all the basic notions of the theory with many examples and several colored illustrations. The author ... provides many informal explanations, several examples and counterexamples to definitions, discussions and warnings about different conventions, and so on. ... It would also make a great read for mathematicians who want to learn about the subject." (Gizem Karaali, MAA Mathematical Sciences Digital Library, January, 2005) "Lie groups are already standard part of graduate mathematics, but their complex nature makes still a challenge to write a good introductory book to it. ... This book is a must for graduate students in mathematics and/or physics." (Arpad Kurusa, Acta Scientiarum Mathematicarum, Vol. 73, 2007) "The book under review therefore makes the wise choice of sticking to linear groups. ... Hall's book has two parts. In the first part, 'General theory', the author introduces matrix Lie groups ... . A highlight of the second part is the discussion of 3 different constructions of irreducible representations of complex semisimple Lie algebras: algebraic (Verma modules), analytic (Weyl character formula), geometric (Borel-Weil construction using the complex structure on the flag manifold). ... this book is a fine addition to the literature ... ." (Alain Valette, Bulletin of the Belgian Mathematical Society, 2009)

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## Table of contents

Preface Part I General Theory 1 Matrix Lie Groups 1.1 Definition of a Matrix Lie Group 1.2 Examples of Matrix Lie Groups 1.3 Compactness 1.4 Connectedness 1.5 Simple Connectedness 1.6 Homomorphisms and Isomorphisms 1.7 (Optional) The Polar Decomposition for $ {SL}(n; {R})$ and $ {SL}(n; {C})$ 1.8 Lie Groups 1.9 Exercises 2 Lie Algebras and the Exponential Mapping 2.1 The Matrix Exponential 2.2 Computing the Exponential of a Matrix 2.3 The Matrix Logarithm 2.4 Further Properties of the Matrix Exponential 2.5 The Lie Algebra of a Matrix Lie Group 2.6 Properties of the Lie Algebra 2.7 The Exponential Mapping 2.8 Lie Algebras 2.9 The Complexification of a Real Lie Algebra 2.10 Exercises 3 The Baker--Campbell--Hausdorff Formula 3.1 The Baker--Campbell--Hausdorff Formula for the Heisenberg Group 3.2 The General Baker--Campbell--Hausdorff Formula 3.3 The Derivative of the Exponential Mapping 3.4 Proof of the Baker--Campbell--Hausdorff Formula 3.5 The Series Form of the Baker--Campbell--Hausdorff Formula 3.6 Lie Algebra Versus Lie Group Homomorphisms 3.7 Covering Groups 3.8 Subgroups and Subalgebras 3.9 Exercises 4 Basic Representation Theory 4.1 Representations 4.2 Why Study Representations? 4.3 Examples of Representations 4.4 The Irreducible Representations of $ {su}(2)$ 4.5 Direct Sums of Representations 4.6 Tensor Products of Representations 4.7 Dual Representations 4.8 Schur's Lemma 4.9 Group Versus Lie Algebra Representations 4.10 Complete Reducibility 4.11 Exercises Part II Semisimple Theory 5 The Representations of $ {SU}(3)$ 5.1 Introduction 5.2 Weights and Roots 5.3 The Theorem of the Highest Weight 5.4 Proof of the Theorem 5.5 An Example: Highest Weight $( 1,1) $ 5.6 The Weyl Group 5.7 Weight Diagrams 5.8 Exercises 6 Semisimple Lie Algebras 6.1 Complete Reducibility and Semisimple Lie Algebras 6.2 Examples of Reductive and

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## Review Text

From the reviews:§§"This is an excellent book. It deserves to, and undoubtedly will, become the standard text for early graduate courses in Lie group theory ... . It is clearly written ... . A reader of this book will be rewarded with an excellent understanding of Lie groups ... . Hall's book appears to be genuinely unique in both the organization of the material and the care in which it is presented. This is an important addition to the textbook literature ... . It is highly recommended." (Mark Hunacek, The Mathematical Gazette, March, 2005)§§"The book is written in a systematic and clear way, each chapter ends with a set of exercises. The book could be valuable for students of mathematics and physics as well as for teachers, for the preparation of courses. It is a nice addition to the existing literature." (EMS-European Mathematical Society Newsletter, September, 2004)§§"This book differs from most of the texts on Lie Groups in one significant aspect. ... it develops the whole theory on matrix Lie groups. This approach ... will be appreciated by those who find differential geometry difficult to understand. ... each of the eight chapters plus appendix A contain a good collection of exercises. ... I believe that the book fills the gap between the numerous popular books on Lie groups ... is a valuable addition to the collection of any mathematician or physicist interested in the subject." (P.K. Smrz, The Australian Mathematical Society Gazette, Vol. 31 (2), 2004)§§"This book addresses Lie groups, Lie algebras, and representation theory. ... the author restricts attention to matrix Lie groups and Lie algebras. This approach keeps the discussion concrete, allows the reader to get to the heart of the subject quickly, and covers all the most interesting examples. ... This book is sure to become a standard textbook for graduate students in mathematics and physics with little or no prior exposure to Lie theory." (L'Enseignement Mathematique, Vol. 49 (3-4), 2003)§§"Though there exist already several excellent text books providing the mathematical basis for all this, introductions aimed at graduate students both in mathematics and physics seem to be rare. So the guiding principle in the planning of the book by Brian Hall ... was to minimize the amount of prerequisites. ... students will benefit from the way the material is presented in this Introduction; for it is elementary and not intimidating, at the same time very systematic, rigorous and modern ... ." (G. Roepstorff, Zentralblatt MATH, Vol. 1026, 2004)§§"This book is a great find for those who want to learn about Lie groups or Lie algebras and basics of their representation theory. It is a well-written text which introduces all the basic notions of the theory with many examples and several colored illustrations. The author ... provides many informal explanations, several examples and counterexamples to definitions, discussions and warnings about different conventions, and so on. ... It would also make a great read for mathematicians who want to learn about the subject." (Gizem Karaali, MAA Mathematical Sciences Digital Library, January, 2005)§§"Lie groups are already standard part of graduate mathematics, but their complex nature makes still a challenge to write a good introductory book to it. ... This book is a must for graduate students in mathematics and/or physics." (Árpád Kurusa, Acta Scientiarum Mathematicarum, Vol. 73, 2007)§§"The book under review therefore makes the wise choice of sticking to linear groups. ... Hall's book has two parts. In the first part, 'General theory', the author introduces matrix Lie groups ... . A highlight of the second part is the discussion of 3 different constructions of irreducible representations of complex semisimple Lie algebras: algebraic (Verma modules), analytic (Weyl character formula), geometric (Borel-Weil construction using the complex structure on the flag manifold). ... this book is a fine addition to the literatu

show more

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