An Introduction to Lie Groups and Lie Algebras (Cambridge Studies in Advanced Mathematics Book 113) 1st Edition by Alexander Kirillov, Jr (PDF)

Description

With roots in the nineteenth century, Lie theory has since found many and varied applications in mathematics and mathematical physics, to the point where it is now regarded as a classical branch of mathematics in its own right. This graduate text focuses on the study of semisimple Lie algebras, developing the necessary theory along the way. The material covered ranges from basic definitions of Lie groups to the classification of finite-dimensional representations of semisimple Lie algebras. Written in an informal style, this is a contemporary introduction to the subject which emphasizes the main concepts of the proofs and outlines the necessary technical details, allowing the material to be conveyed concisely. Based on a lecture course given by the author at the State University of New York, Stony Brook, the book includes numerous exercises and worked examples, and is ideal for graduate courses on Lie groups and Lie algebras.

User’s Reviews

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

⭐It is trying too hard to be a rigorous introduction without mentioning the differential topology behind everything. It is clear that it assumes a background in differential topology but every time a proof involves stepping in that direction it is omitted.7 When you get to the exercises there are many that would be good, albeit very long, if there was more differential topology. It is fine to sweep many of the rigorous technical details under the rug, and needed to get to the juicy part of the subject, but craft the exercises to avoid such details as well. Once the book switches to Lie algebras and representations it does clean up its act a bit. There is also a glaring omission of maximal tori (or Borel subgroups) which are essential in any further study.

⭐Kirillov’s coverage is 1/2 Lie Groups and 1/2 Lie Algebras. It occupies a unique middle ground between books which rush through Lie Groups in their haste to get to Lie Algebras, and more formal mathematical treatments. I personally feel Kirillov negotiated this quite well, injecting essential insight where needed but never bogging the book down in detailed proofs of major theorems. I personally found the problems both interesting and helpful. I would recommend the book to those who’ve encountered Lie Groups and want to build a solid intuition. On the other hand, those looking for yellow books filled with unmotivated theorems and proofs will be disappointed.

⭐I used this book as the primary text for an introductory course on Lie groups and Lie algebras. There are several aspects of the book which distinguish it from every other book on the same topic, making it an indespensable resource for the beginning student.First, the book is, as its title indicates, an introduction, and a fairly brief one at that. It is not intended to be comprehensive in scope or in depth, rather to gently introduce some fairly complex ideas in the most basic way possible. This is the primary reason it is so useful to start with: The author knows just how much detail is necessary and skips cumbersome and unenlightening proofs. For example, he doesn’t prove Serre’s theorem or finish the proof of the PBW theorem, but rather refers to other books for these. In contrast to other books on the subject, the student doesn’t have to sift the important points from the nitty-gritty details. Every section is important and worth reading. I particularly appreciate that the sections on Lie groups don’t require that the reader is an expert in differential geometry and reviews all essential prerequisites.Of particular value is the excellent collection of exercises. The majority of these are not particularly difficult, but most are enormously worthwhile. Having done lots of exercises from other books, including Knapp (Lie groups beyond an introduction), Hall, Humphreys, and others, I can safely say these are among the best, reaching both an optimal level of difficulty and a fair balance between computation and theory. (Note: Hall’s book has great exercises too and are good for those who want more practice with computations). One of the main problems I have with problems in most math books is that they often feel unrelated to the material of the book and don’t help to understand the material. Kirillov’s exercise practically all require verifying simple details from the book or proving small parts of theorems and are all worth doing.Finally, the book outlines many more advanced directions in Lie theory and gives appropriate references. Overall, the book has the feel of a rigorous exposition without scaring away the student with 800 pages of technical details. Of course there are simpler texts (like Hall) which just focus on matrix Lie groups and more sophisticated (Knapp) which contain everything in this book and a LOT more, but I’d say for a first read, this book is the most suitable.

⭐this is a wickedly good book. it’s concise (yeah!) and it’s well written. it misses out on lots of stuff (spin representations, etc..). but once you read this book you will have the formalism down pat, and then everything else becomes easy.if you put in the hours to read this book cover to cover — like sitting down for 3 days straight 8 hours a day, then will learn the stuff. if you don’t persevere and get overwhelmed with the stuff that is not clear at the beginning, then you will probably chuck it out the window.lie groups and lie algebras in 200 pages done in an elegant way that doesn’t look like lecture notes cobbled together is pretty impressive.

⭐Un désastre didactique, vide de concept et de mise en perspective. Fuyez

⭐

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