# Rigorous Lie Group and Lie Algebra Textbooks for Physicists

Hi everyone,

I was just wondering if anyone had any suggestions of more-mathematically-rigorous textbooks on Lie groups and Lie algebras for (high-energy) physicists than, say, Howard Georgi's book.

I have been eying books such as "Symmetries, Lie Algebras And Representations: A Graduate Course For Physicists" by J. Fuchs and C. Schweigert and "Lie Groups, Lie Algebras, and Some of Their Applications'' by R. Gilmore; however, the problem that arises with those, and with pure mathematical books on the subject, is that their (exponential map) convention is different from that used in physics: in physics, the Lie algebra of a matrix Lie group, G, is defined to be the set of all matrices X such that exp[itX] is in G for all real numbers t, whereas in math, the Lie algebra of G is defined by exp[tX], without the i. This leads to different conventions/results throughout the entire subject (for instance, the Lie algebra of the unitary group is the space of all hermitian matrices in physics, but the space of all ANTI-hermitian matrices in math -- it is obvious why the former convention is chosen in physics).

Even though the two textbooks that I listed above are meant for physicists, they have adopted the mathematical convention (I guess that is in line with their intent to be more mathematically rigorous), so I would like to look elsewhere, if possible.

So, just to summarize (since this post has become rather long -- sorry about that!): I am looking for a mathematically-rigorous textbook on Lie groups and Lie algebras that uses physicists' conventions.

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I guess there's always Schlomo Sternberg https://www.amazon.com/dp/0521558859/?tag=pfamazon01-20&tag=pfamazon01-20

But he can be somewhat long-winded and arrives at results in a somewhat round-about way. His proof of SO(3) irreps being labelled by integer "spins" (for lack of a better term) is what made me notice his style. It's a little grating, but otherwise it's a decent book.

There's also Cornwell, but I despise his style. And his notation is awful to read. He has a 3 volume series that's really detailed, and a book that consolidates the first two volumes. I would go with the first two volumes as opposed to the consolidated book. Just my opinion.

edit: though to be honest i don't know what's especially wrong with Georgi. I think he's plenty rigorous, it's just that most of the stuff that requires proving is done so fairly easily through contradiction.

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Thanks, naele, for your reply! I'll have to check out the Sternberg book, but I have gotten the Cornwell books from the library, and I'll have to agree with you -- his notation and type-setting is not the easiest on the eyes. He also, unfortunately, uses the exp[tX] convention. Maybe that's just something that I'll have to learn how to deal with, I suppose?

Don't get me wrong: I think Georgi's book is great! But he's not really rigorous in that his progression through the textbook is more computational and example-based, rather than proof-based. Also, he doesn't cover Lie groups, and, in addition to studying representation theory of Lie groups and Lie algebras, I'm interested in delving more deeply into the connections between Lie groups and Lie algebras (exponentiation, one-parameter subgroups, connectedness, infinitesimal forms / generators, etc.).

So it seems unavoidable that if I want to pursue that, it has to be with the math conventions rather than physics conventions -- unless anyone is able to come to the rescue?!

Somebody can correct me on this, but using exp[tX] doesn't matter if you've defined your Lie algebra to use Hermitian generators. Just take X-> iY or whatever and you're set. They should give you the corresponding group element (try with su(2) -> SU(2)).

You're absolutely correct; applying that transformation to all results (e.g., commutation relations, structure constants, etc.) reconciles the two differing conventions. I guess I was just trying to see if there was any way I could be a bit lazier when reading through a textbook on the material, but I guess a bit of extra work will keep the mind sharp! :)

Thank you for all of your help!

If it's any comfort, the exponential maps isn't used all that much after it's introduced and discussed, mainly because physicists are more interested in the representations.

You might be interested in a book like https://www.amazon.com/dp/0387985794/?tag=pfamazon01-20&tag=pfamazon01-20 or https://www.amazon.com/dp/0521829607/?tag=pfamazon01-20&tag=pfamazon01-20 which are quite rigorous, though not especially complete. It might not be what you're looking for, since the chapters on Lie groups/algebras requires understanding of the previous chapters on differential geometry/manifolds. But that's the more abstract definition of a Lie group which is not nearly as useful to a physicist as just thinking about the matrix Lie groups.