# References and suggestions about the elementary theory of Lie groups and Lie algebras

I am looking for suggestions on how to approach the field of Lie groups and Lie algebras. I am acquainted with both the elementary algebraic concepts, having studied from Bourbaki's "Algebra I-III", and the elementary theory of manifolds, both topological and smooth, having studied from Lee's "Introduction to Topological Manifolds" and "Introduction to Smooth Manifolds". Lee actually devotes some chapters to Lie groups and Lie algebras, which I have read, but I somehow feel that studying from textbooks which are aimed at specifically covering Lie groups and Lie algebra would be a better starting point, even from an introductory point of view.

So, where do I start from? My interest is mostly in the applications of Lie theory to theoretical Physics, but I like to approach mathematical topics from a mathematician's point of view, so that I don't lose anything on the way. First of all, I would like to know how the field of Lie groups and Lie algebras is structured, i.e. what are the aims of the theory, its main contents and so on, so as to be able to know my way around, so to say. Then, I would like to know what textbooks you think are good introductions to the subject. As I said, I'm acquainted with the theories of smooth manifolds and algebras, so, if possible, I'd appreciate recommendations about books that skipped those topics. On the other hand, I haven't yet studied anything about complex manifolds, so I'm wondering whether I should have a good look at that subject first. The topics I'm especially interested in are Lie groups representations, structure constants of the Lie algebras and Lie group actions on smooth manifolds. I would also need some covering of semi-direct products of Lie groups and especially of their Lie algebras (read: I must know how to cope with the Poincaré group). Finally, I'd appreciate recommendations on textbooks that explicitly cover the applications of Lie theory to theoretical Physics: quantum mechanics and particle physics, general relativity, hamiltonian systems (preferably with some covering of the Lagrangian approach to mechanics and of Noether's theorem). Thank you in advance for your answers.

P.S.: I own a copy of Bourbaki's "Lie Groups and Lie Algebras I-III". How can I make it fit in my study?

I'm interested in the same fields.

I liked very much the approach to the Lie theory of Anthony Knapp - Lie Groups Beyond an Introduction. Here the author gives a very beautifull and complete overview on the world of Lie Groups and Lie Alebras. Starting from the topological description, looking at a lot of concrete examples, he introduces lots of deep results in Representation Theory and about Classification of Lie Groups and Lie Algebras, looking very often to the strength link between this structures (and their link with topology)

About Representation of Lie Algebras a very good book is J. Humphreys - Introduction to Lie Algebras and Representation Theory. Starts from the basic thery of Lie Algebras and, using a lot of Linear Algebra, proof some deep results about Structure, Representations and Characters of this very improtant objects.

If you like combinatorics, you can be interesten to the theory of Representation you can look at Fulton, Harris - Representation Theory. There are a lot of chapters about the Representations of Lie Algebras.

Another interesting book could be V.S.Varadarajan - Lie Groups, Lie Algebras and Their Representation for this Theory viewed by an analytic approach.