# Getting started with Lie Groups

I am looking for some material (e.g. references, books, notes) to get started with Lie Groups and Lie Algebra.

My motivation is that I (eventually) want to understand the theory underpinning papers such as these.

The problem is, I am at the Rumsfeldian stage where I don't know what I don' t know. The wikipedia page makes it clear that I lack several prerequisites in modern geometry.

So, what would be a proposed study path to get to this stage? My background is electronic engineering and am comfortable with associated topics (signal processing, estimation theory, Kalman Filtering, etc.)

• "Integrating generic sensor fusion algorithms with sound state representations through encapsulation of manifolds" ...snarXiv candidate right there. Aug 6 '13 at 11:48
• Sep 23 '14 at 4:54

I would suggest you start with chapter 4 of An Introduction to Manifolds by Tu, Then study Lie Groups, Lie Algebras, and Representations: An Elementary Introduction by Hall and finally study Differential Geometry, Lie Groups, and Symmetric Spaces by Helgason.

Good luck!

• +1 and the recommendation of Helgason's textbook is great! Aug 6 '13 at 12:34
• All answers have been useful, but this answer is the most complete. Thank you. Aug 8 '13 at 22:59
• @Damien: Except it will probably take you years instead of months to understand those papers if you follow this route. The amount of Lie theory one needs to understand to grok those applications/papers you've linked can be mostly served from an undergraduate textbook on Lie theory.
– Fizz
Apr 12 '15 at 14:18

Stillwell: Naive Lie Theory is a great first introduction since it covers the very basics and uses SO(3) and SU(2) as examples. And it gives a sneak-preview of group theory and topology too. Though, more advanced topics such as adjoint representation and left-invariance are not covered.

The book of Zhelobenko is written with applications in view . It has a whole chapter on $SU(2)$ (which is isomorphic with $SO(3)$), so it seems to fit with the paper you're interested in.

• +1 and nice recommendation! I've never heard of this textbook but it looks excellent. Aug 6 '13 at 12:16
• Note that SU(2) is not isomorphic to SO(3). It is, however, a a covering group of SO(3), so SU(2) and SO(3) are definately intimately related. Aug 3 '18 at 22:45

I'd say this book: http://www.amazon.com/Lie-Groups-Algebras-Representations-Introduction/dp/0387401229 would be a great choice given the information that you've provided. The link I've provided is to the Amazon webpage for the textbook.

If you don't have access to a library or can't buy the book, then: http://arxiv.org/pdf/math-ph/0005032v1.pdf is an link to an abridged version of the textbook (?) on the arxiv. So, all you need to do is to click on the link and you've got legal access to a nice introduction to Lie groups and Lie algebras. It starts right from the beginning so prerequisites are pretty much minimal.

Disclaimer: I haven't read the textbook but from all I've heard it's great (and I did tutor a course for which this book was the official text so I know the exercises are great). Personally, I learnt what I know of the subject from the book Lie groups by Daniel Bump but I think that's more advanced (though by all means go for it if you think you're up for it!).

I hope this helps!

• +1 for arxiv version. It's the second recommendation for Hall, so I will have to take a good read of it. Aug 8 '13 at 23:00

The book by George W. Bluman contains elegant introduction to Lie groups with large number of examples and applications.