# Learning Lie Groups and Manifolds: An Easy Guide with Examples

• starfield
In summary, there is no one introductory book on Lie groups and manifolds, but rather a collection of intermediate level texts that can be pieced together to create a comprehensive understanding of the subject. Some recommended texts include Artin's "Linear Groups," Warner's "Foundations of Differentiable Manifolds and Lie Groups," Rossman's "Lie Groups: An Introduction through Linear Groups," Chevalley's "Theory of Lie Groups," Knapp's "Lie Groups Beyond an Introduction," Bump's "Lie Groups," and Helgason's "Differential Geometry, Lie Groups, and Symmetric Spaces." For physicists, Frankel's "The Geometry of Physics" and Nakahara's "Geometry, Topology, and Physics"
starfield
hi all!
could anyone suggest a good text for learning lie groups( and other continuous groups) and manifolds...something that has a lot of solved examples(since i'll be learning the subject on my own) ...something of the sort of the text on algebra by fraleigh(which has a lot of examples and is easy reading,yet not boring)...

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What kind of applications are u interested in ?Physics offers plenty of examples...That is to say,are u looking for a book for a mathematician,or for a physicist?

Daniel.

the absolute beginning reference is a few chapters of mike artins algebra book, called linear groups. it is his sophomore abstract algebra book from MIT. it assumes nothing and introduces the basic examples of SO(3) and SU(2), and the standard map between them.

hi
i saw the stuff in artin's book...but he doesn't give much examples, isn't it? i want it from a mathematician's point oif view...

well artin is certainly a mathematician, so he is giving a very small but important set of examples from a mathematician's point of view. i recommend reading artin first carefully. that pair of exampels, SO(3) and SU(2) and the souble cover between them will teach you a lot already.

then you might look at Frank Warner's book Foundations of differentiable manifolds and lie groups, or W. Rossman's Lie grpups, and introduction through linear groups, or the classic book by Chevalley, on Theory of Lie Groups.

I myself am not an expert. Indeed lie groups is probbaly one of the biggest gaps in my knowledge of classical mathematics.

The Rossmann book is a good introduction. Another good one is the recent one by B.Hall, published by Springer. For introductory grad level treatments you might like to take a look at Knapp's Lie Groups Beyond An Introduction (Birkhauser), and the recent book by Bump, simply titled Lie Groups. The old text by Helgason, titled Differential Geometry, Lie Groups, and Symmetric Spaces, is also a good read. Warner is another fine text.

The physicists I've met have had problems with math texts. The books above are written for math students, and they assume a way of approaching the subject that is alien to physicists: careful definitions, elegant proofs, and a structure of results piled one on another. Physicists tend to think heuristically (well, so do mathematicians, but they can translate heuristic insights into abstract definitions, carefully-stated theorems, and formal proofs). Also, the grad level texts often assume you can construct your own examples, and develop heuristic insights from formal treatments by yourself.

A physicist would be better off with books such as Frankel's The Geometry of Physics, or Nakahara's Geometry, Topology, and Physics.

For calculations, I think there's an old text by Curtis(?) titled Matrix Groups, published by Springer. Another good one is Sattinger and Weaver's Lie Groups and Algebras, with Applications to Physics, Geometry, and Mechanics, also published by Springer.

By the way, I assume you've got a decent background in real analysis, linear algebra and general topology. And I'll repeat what mathwonk has already suggested: work carefully through the relevant sections of Artin: he develops baby Lie theory well.

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that is a very insighful and well written post, bombadillo. I enjoyed the analysis of the contrasts between mathematicians' and physicists' thinking and learning and writing styles. glad to have you around. i am sure your book recommendations are valuable as well.

I've spent a lot of time studying Lie theory on my own, and I will have to disappoint you by saying that there are no beginning books on Lie theory comparable to Fraleigh's. The literature on Lie theory begins at an intermediate level. You will have to piece your knowledge together from the first few chapters of more advanced works, and from online references. Knapp's book is outstanding - it treats abstract Lie theory, the formulations are very clear, it has excellent examples, and discusses areas of Lie theory that are rarely mentioned outside of obscure journals, such as Vogan diagrams. Sattinger and Weaver's book is again excellent, but it treats Lie theory from a much different perspective, from the standpoint solving differential equations, which is where it actually originated. Frankel's book is mostly from this perspective as well - if you don't already have this book go buy, it has so much in it, every time I read it I find something deep that I never noticed before. I hated Chevalley's book, but many have found it useful, and it can be gotten for a few bucks, so it is at least worth checking out. Some other books I have found to be valuable are:

Dictionary on Lie Groups and Algebras, Frappat et al.
Lie Algebras, Jacobson
The Classical Groups, Weyl
Group Theory and its Application to Physical Problems, Hammermesh
Lie Algebra in Partical Physics, Georgi
Geometrical Methods of Mathematical Physics, Schutz

There is a excellent, though short, appendix on Lie theory in Michio Kaku's Introduction to Superstrings and M-theory.

to me this again confirms the value of artin's chapters on linear groups, as they DO begin at an introductory level.

lie groups are an abstraction of the properties of matrix groups, so that is the place to begin. artin begins with the absolutely most basic examples, and explains them completely. so that is the place to start. he himself says afterwards that he cannot think of any books to recommend as follow up, since there are no suitably elementary continuations of his discussion.

hi all!
thanks a lot..! now i have the names of a lot of books...some place to begin with...

i'll start with artin's book, i guess...that seems the most elementary and easily available book...

The good thing is that linear operators, clifford algebras and spinors are powerful enough to compensate over differential geometry. It all depends on the approach one wants to take.

The best way is to begin with some stuff not directly related to Lie groups:

1) vector fields on manifolds (coordinate and coofdinate-fre understanding of them);
2) systems of first order ODE's associated with the vector fields, their solutions, and coordinate-free interpretation of these solutions as curves;
3) one-parametric transformations associated with the vector fields and conversely vector fields asociated with one-parametric transformation;
4) commutator of the vector fields (in connection with the commutators of one-parametric transformations);
5) smooth mappings of manifolds to other manifolds and transportation of vectors under such mappings;
6) Frobenius theorem on involutive distributions.

If the above 6 items are firmly understood, the basics of Lie group theory look like a specialization of the above things for manifolds with an additional algebraic structure.

As to me, the examples of matrix groups GL, SL, SO, SU make more confusion than clarify anything at the beginning stage of learning.

you may be right for a certain audience, but of course you realize you are recommending exactly the opposite of what he asked for, since he wanted an example - oriented approach.

the approach you suggest is approximately the content of frank warners book, but i myself find it odd to recommend someone "start" with a months long or years long program of learning manifolds and differential geometry rather than seeing a few examples of the most usual and useful lie groups.

just goes to show how many different learning styles there are.

mathwonk said:
but i myself find it odd to recommend someone "start" with a months long or years long program

I used Kobayashi & Nomizu's book, the first 2 sectioins in their Vol. 1 of Foundations of Dif. Geom. It's rather difficult book in whole, but the first 2 sections make about 20 pages, so it's not so much. There are almost no proofs there or very short and not clear proofs, but the statements and layout are good for making your own proofs (as an exercise). I want to say that a bad book is sometimes better than a good detailed book for not to follow the author's ("months and years long") way, but choose your own.

As for Lie groups, the most tricky thing here is the passage from the multiplicative commutator A*B*A^{-1}*B^{-1} to the half-additive commutator a*b-b*a. The matrix structure itself in GL and in other matrix groups casts a shadow over this passage. Dealing with matrices here it would be better to understand them as one-dimensional arrays of N^2 numbers and treat the marix multiplication as some tricky operation over these arrays obeying the associativity.

I think it depends from which side one approaches the subject (QFT or relativity). Lie groups can also be covered with the methods of group theory and geometric algebra. Functional analysis is also very helpful.

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## 1. What is the purpose of learning Lie groups and manifolds?

The purpose of learning Lie groups and manifolds is to understand the mathematical structures that describe symmetries and transformations in various fields, such as physics, engineering, and computer graphics. These concepts are essential in many areas of science and can provide a deeper understanding of complex systems.

## 2. What are some real-world applications of Lie groups and manifolds?

Lie groups and manifolds have a wide range of applications, including in robotics, computer vision, data analysis, and differential geometry. They are also used in the study of physical systems, such as quantum mechanics, relativity, and fluid dynamics.

## 3. Is prior knowledge of advanced mathematics necessary to learn Lie groups and manifolds?

While some familiarity with linear algebra and calculus is helpful, it is not necessary to have a deep understanding of advanced mathematics to learn about Lie groups and manifolds. A basic understanding of algebra and geometry is sufficient to grasp the concepts and examples presented in this guide.

## 4. How can I apply the concepts learned in this guide to my own research?

The examples and exercises in this guide are designed to provide a foundation for understanding Lie groups and manifolds and their applications. You can apply these concepts to your own research by identifying and utilizing symmetries and transformations in your data or system of interest.

## 5. Are there any additional resources for further learning about Lie groups and manifolds?

Yes, there are many resources available for further learning about Lie groups and manifolds, including textbooks, online courses, and research papers. It is also helpful to practice applying these concepts to real-world problems and to seek guidance from experts in the field.

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