Lie Groups, Lie Algebras, and Some of Their Applications (Dover Books on Mathematics) by Robert Gilmore (PDF)

Description

Lie group theory plays an increasingly important role in modern physical theories. Many of its calculations remain fundamentally unchanged from one field of physics to another, altering only in terms of symbols and the language. Using the theory of Lie groups as a unifying vehicle, concepts and results from several fields of physics can be expressed in an extremely economical way. With rigor and clarity, this text introduces upper-level undergraduate students to Lie group theory and its physical applications.An opening discussion of introductory concepts leads to explorations of the classical groups, continuous groups and Lie groups, and Lie groups and Lie algebras. Some simple but illuminating examples are followed by examinations of classical algebras, Lie algebras and root spaces, root spaces and Dynkin diagrams, real forms, and contractions and expansions. Reinforced by numerous exercises, solved problems, and figures, the text concludes with a bibliography and indexes.

User’s Reviews

Reviews from Amazon users which were colected at the time this book was published on the website:

⭐well written and authoratative on subject of Lie groups. Easy to understand. unusual for a math book. Youtube videos by XylyXylyX on Lie Groups follow this book. Used in combination, you may learn Lie Groups yet!!

⭐It’s a solid book. It covers both mathematical subtleties and physical applications. It’s written at a very advanced level.However, a warning to the uninitiated: He is sloppy with his notation. It’s rarely explained or made precise. You sort of have to unravel what the hell he is trying to say. For example, he heavily relies on Einsteinian notation in his presentation on covariance, contravariance and metrics, but I don’t think he ever mentions he is using it. If you are not already very familiar with the types of conventions that he chooses to use, you will find that the notation quickly interferes with grasping the material. I often found that the hardest part of understanding his explanations was understanding how he was using notation. I understand that he is using professional language, but when that becomes the biggest obstacle in understanding the material, perhaps there is a better way of doing it. In this sense, I wish he was more of a mathematician.

⭐Simply put, I didn’t like the book. It is too ambitious and this is bad for the reader, that can easily get lost amidst too many irrelevant details.The author himself states in the preface of his newer book (R. Gilmore, ”

⭐”) that “Over the course of the years I realized that more than 90% of the most useful material in that book [the one being revised here] could be presented in less than 10% of the space.” What else can I say?The application-minded readers (e.g., physicists) will suffer from its style and contents, and the mathematician can find much better presentations elsewhere in the vast literature on the subject. So… who needs it? I enjoyed and profited much more from the book by B. G. Wybourne, ”

⭐.”

⭐I’m really enjoying this text – several of my math books, as much as I love them, are so dense that at times I feel like I’m slogging through them. What I really appreciate about this book are the excellent figures and comprehensive summary tables; these are thoughtfully made and help solidify concepts and/or “big picture” ideas. I generally avoid books with too many words (I find that my style of learning leans towards equations, pictures, worked examples, and efficiently stated insights), and this text strikes just the right balance.

⭐Surely the subject matter of Lie Algebras must be more than difficult if there exist Schaum Outlines even about Tensor Theory (see e.g. Kay, Tensor Calculus), but NOT about Lie Algebras. Nevertheless, Gilmore manages to mess up things quite completely.Besides, what’s the point of requesting such an inordinate amount of prerequisites? It’s like saying: “I’ll explain you Differential Geometry, but you must already know Quantum Physics, Lagrangian Mechanics, Algebraic Topology, Tensor Theory, Measure Theory, General Relativity and Cosmology” (so to speak). Thanks, prof. Gilmore, had I had such a background, I’d teach YOU how to write a DECENT paper on Lie Algebras.Being not yet convinced, you should compare Gilmore’s rambling exposition with a gem of graduate-level divulgative literature (although concerning the different topic of functional analysis): Infinite Dimensional Analysis, A Hitchhiker’s guide, of Aliprantis and Border. Over 600 crystal-clear and extremely rigorous pages teaching everything you need to know about Measure Theory, Advanced Topology, Riesz Spaces and much more.Gilmore’s book has a number of unpleasant features, among which the constant mixing up of the complex case with the real one. Exposition jumps ceaselessly from linear vector functions on the real field to sesquilinear functions on the complex field, from orthogonal matrices to unitary matrices and so on and on.Here are some precious samples picked from the first 100 pages:Page 28: tensor notation (the ill-famed tensor product symbol “⊗”) is abruptly introduced without any previous explanation, together with such inspired mystical definitions as “a tensor is a vector”. Gilmore then goes on happily this way for three more pages, with the result you might imagine.Page 69: a (n)hasty exposition of “Some realization for a continuous group of transformations”, barely a page long, leaving more doubts than elucidations.Page 78: The author begins speaking of Sigma-Algebras and μ-Integrals, terms typical of measure theory, without any previous warning. You baffled? Come on! Isn’t it true that even children know about measure theory, Lebesgue integrals, Sigma Algebras and Measurable Functions?That was enough for me. I didn’t read farther. As soon as I reached page 100 I gave up. Next thing I ran a search of a reimboursement clause throughout the book. Too bad I found none. I could have started a lawsuit. Next after that, I bought online the excellent book of Hall, Lie Groups, Lie Algebras and Representations, edited by Springer-Verlag, that I am currently and profitably reading.Thanks, prof. Gilmore, for a waste of money (29,95 USD) and several tens of unprofitable hours (plus thirty minutes to write and post this review).

⭐I haven’t read this whole book cover to cover, because of time constraints. However, I can say that it is extremely clear in it’s exposition. The material is very well chosen for use by physicists. I have read pure math books on this topic, and while they can be more sophisticated and thorough, they are rarely as straight forward, nor do they cover the breadth of material in this book.In sum I would have to agree with what I was told: “this is the book on Lie Algebra for a physicist”.

⭐This book is not worth the paper it is printed on,.it has an impressive title. It is disorganized and unreadable on account of the awful notation the author uses throughout to I regret that I bought it and I would like to return it. Please,. let me know how I can do that, since I still have the envelope that it came in.I hope to hear from you soon.

⭐If you plan on reading this on a Kindle, forget it. The equations are rendered in a tiny font, much smaller than the text, and in grey instead of black, and enlarging the text has no effect on the equations. It is barely possible to make out the equations by rotating the screen to landscape view. I tried it on Kindle for PC, same problem. The publisher should hang their head in shame for turning out such a pathetically shoddy product.

⭐Dieses Buch erklärt und probiert viele wichtige Theoreme der Gruppentheorie durch nicht komplizierten Beispiele und macht alles so leicht wie möglich. In diesem Buch findet man auch die Lies Theoreme und ihre wechselseitige, die schwer zu finden sind. Dieses Buch ist ein Hinweis, der in vielen Büchern über Gruppentheorie als Literatur empfohlen wird.This is a good introduction for those of you who are a bit “mathphobic” but still want to understand the “difficult” topics and techniques in mathematics.

⭐V good

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