0

I am currently looking for an introductory text on Lie groups and Lie algebras with an emphasis on differential geometry. (I already have a little bit of background knowledge on smooth manifolds, having read through a fair amount of John M. Lee's Introduction to Smooth Manifolds.) After some searching online I found these three textbooks:

  • Sigurdur Helgason, Differential Geometry, Lie Groups, and Symmetric Spaces
  • V. S. Varadarajan, Lie Groups, Lie Algebras, and Their Representations
  • Anthony W. Knapp, Lie Groups Beyond an Introduction

I've heard that Knapp may be too advanced for an introductory textbook. I would also be open to any other suggestions.

Hrhm
  • 3,303
  • 2
    Knapp's book is not too advanced, but the other books have more "emphasis on differential geometry". I can recommend Warner's book Foundations of Differentiable Manifolds and Lie Groups. Many books on Lie groups try to bring the problems to the level of Lie algebras, where we can solve them with linear algebra and vector spaces. This is "easier" than differential geometry (of course this is no option for you, as you said explicitly in your title). – Dietrich Burde Sep 19 '22 at 15:12
  • Here are the opinions of some other MSE users: this post,and this post. Quote (from Qiaochu Yuan):"You don't need to know any differential geometry to grasp the basic ideas in Lie theory beyond some idea of what a tangent vector is. The study of semisimple Lie groups is largely algebraic and getting a good grasp of the important examples doesn't require more than comfort with calculus and linear algebra." – Dietrich Burde Sep 19 '22 at 15:15
  • Correction to my previous comment: As an alternative to Frank Warner's excellent book, I recommend Bröcker and Tom Dieck's Representations of Compact Lie groups. If I may add, in my opinion Daniel Bump's book is a terrible reference with lots of mistakes in the text and the exercises. I had confused Knapp with Bump. Apologies to Knapp, a book of which I have heard pretty good reviews. – Laz Sep 19 '22 at 15:21
  • Overall I agree with Qiaochu Yuan's take, although as far as I've seen in every approach I have read, you need to understand flows on manifolds and at least one of the versions of Frobenius Integrability Theorem. – Laz Sep 19 '22 at 15:26
  • 1
    There’s no differential geometry in Varadarajan. There’s plenty in Helgason. – Ted Shifrin Sep 19 '22 at 15:36
  • @DietrichBurde Thank you for your helpful replies! I think I maybe overemphasized differential geometry in my post. I mainly want a textbook which handles Lie groups in terms of manifolds, as opposed to (for example) Hall's Lie Groups, Lie Algebras, and Representations, which only deals with matrix Lie groups. – Hrhm Sep 19 '22 at 17:22
  • Warner's book is great, but overly abstract. A book that, based on its table of contents, looks good is https://link.springer.com/book/10.1007/978-1-4471-0183-3#toc – Deane Sep 19 '22 at 17:22

0 Answers0