Lie Groups, Lie Algebras, and Representations: An Elementary Introduction - Graduate Texts in Mathematics 2nd ed. 2015 edition

Description for Sales People: This book provides an introduction to Lie groups, Lie algebras, and representation theory, aimed at graduate studnets in mathematics and physics. Review Quotes: 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 pReview Quotes: 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 exTable 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 andPublisher Marketing: 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 allows the reader to get to the heart of the subject quickly, and covers all of the most interesting examples.

Contributor Bio:  Hall, Brian Besides Brian Hall's latest work, Madeleine's World, his previous writings include two novels, The Saskiad and The Dreamers; two nonfiction books, Stealing from a Deep Place: Travels in Southeastern Europe and the Impossible Country: A Journey Through the Last Days of Yugoslavia; and articles for the New Yorker, the New York Times Magazine, and Granta. Hall currently lives in Ithaca, New York.