Lie Groups, Lie Algebras, and Representations

This book provides an introduction to Lie groups, Lie algebras, and repre­ sentation theory, aimed at graduate students in mathematics and physics. Although there are already several excellent books that cover many of the same topics, this book has two distinctive features that I hope will make it a useful addition to the literature. First, it treats Lie groups (not just Lie alge­ bras) in a way that minimizes the amount of manifold theory needed. Thus, I neither assume a prior course on differentiable manifolds nor provide a con­ densed such course in the beginning chapters. Second, this book provides a gentle introduction to the machinery of semi simple groups and Lie algebras by treating the representation theory of SU(2) and SU(3) in detail before going to the general case. This allows the reader to see roots, weights, and the Weyl group "in action" in simple cases before confronting the general theory. The standard books on Lie theory begin immediately with the general case: a smooth manifold that is also a group. The Lie algebra is then defined as the space of left-invariant vector fields and the exponential mapping is defined in terms of the flow along such vector fields. This approach is undoubtedly the right one in the long run, but it is rather abstract for a reader encountering such things for the first time.

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’EnseignementMathematique, 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 theflag manifold). … this book is a fine addition to the literature … .” (Alain Valette, Bulletin of the Belgian Mathematical Society, 2009)