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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?

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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.

About Lie Theory and Phisic, I suggest to read this:

Applications of Algebra in Physics

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  • $\begingroup$ Do any of them treat complex Lie groups (complex as in complex numbers)? I totally forgot to ask. Were they good tools to better understand the connection between Lie groups and theoretical physics? $\endgroup$ – Giorgio Comitini Jun 29 '15 at 16:17
  • $\begingroup$ Oh yes, I think complex lie groups are deeplely debated in every book about Lie Theroy. About the link with Teoretical phisic, in an answer to my question there are serveral links to notes about this argument. But I've to be honest: I feel as this link is not well explored by Mathematicians... $\endgroup$ – Sabino Di Trani Jun 29 '15 at 16:29
  • $\begingroup$ That is what I feared. Although, as long as I have references about Physics topics that should be enough. Thank you for the suggestions, I'm having a look at Knapp's book. $\endgroup$ – Giorgio Comitini Jun 29 '15 at 16:36
  • $\begingroup$ Hi Joseph. I found Knapp's book at my University's library. Here Knapp says that a prerequisite for the book is an acquaintance with the elementary Lie Theory. Do you know what he means exactly? I currently know what a Lie group and a Lie algebra are, and very few other notions (I would not regard it as an acquaintance with elementary Lie Theory). Do you think I should start with some other book, or do you know if, for example, Lee's treatment of Lie groups/algebras is enough? $\endgroup$ – Giorgio Comitini Jun 30 '15 at 12:22
  • $\begingroup$ I studied Knapp with no great basics in Lie Theory. I think that definitions and basics facts about the differential structure can be sufficient. Probabily the only very important result required (just the term of the theorem, it's differential geometry) is the theorem of correspondence between Lie Sub-Algebras and Lie SubGoups, but is a boring (don't kill me differential geometers!) theorem: you can find it in every book od DiffGeom. $\endgroup$ – Sabino Di Trani Jun 30 '15 at 12:38

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