Wikipedia:Reference desk/Archives/Mathematics/2007 July 15

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July 15

Group Theory Textbook Recommendations Wanted

Does anyone have recommendations for a textbook or sequence of textbooks that would work well for independent study of Lie groups? I'm particularly interested in a thorough understanding of the inhomogeneous Lorentz group and its subgroups, especially SU(2).

I've gotten started studying Hamermesh's Group Theory and its Application to Physical Problems, which looks like it should cover most of what I want to learn. But the book was written 45 year ago, and it's rather light on problem sets, so it seems like there must be something better out there.

As far as background assumptions, I'm knowledgeable about elementary quantum mechanics, relativity, and some basic group theory. But if the book(s) started from the beginning on group theory, that would be good, as I could use the review. MrRedact 02:24, 15 July 2007 (UTC)

You might want to take a look at Wu-Ki Tung, "Group Theory in Physics", ISBN-13: 978-9971966577. I'm not sure if it studies Lie groups, though. --Waldsen 04:36, 15 July 2007 (UTC)

• Abstract Theory of Groups - O. Schmidt
• An Elementary Introduction to Groups and Representations - B. Hall
• Group Theory (Lie's, Tracks and Exceptional Groups) - P. Cvitanovic
• Group Theory Exceptional Lie Groups As Invariance Groups - P. Cvitanovic
• Introduction to the Theory of Groups - G. Polites
• The Theory of Groups - H. Bechtell
• the Theory of Groups 2nd ed. Vol.1 - A. Kurosh
• Theory Of Groups of Finite Order - W. Burnside

--Cronholm¹⁴⁴ 04:46, 15 July 2007 (UTC)

Since you express special interest in SU(2), we should point out that it is isomorphic to the group of unit quaternions, which is also the unit 3-sphere, S³, one of the handful of spheres that admit a group structure. It is one of the most accessible examples of a Lie group, with lots of easily visualized algebra, geometry, and topology. Rather than start with Lorentz and specialize, try getting a grip on this one example and generalize.

A number of abstract algebra books cover the basics of group theory; as one example, I like Mac Lane and Birkhoff, ISBN 978-0-8218-1646-2 (not the same as Birkhoff & MacLane, ISBN 978-1-56881-068-3); one plus is that it gently eases in a little category theory. Almost any introduction to groups will lean heavily towards finite groups. Step two, just getting comfortable with Lie groups for the first time, I'll have to think about. Meanwhile, I'd suggest brushing up on topology. But the next step should probably include Fulton & Harris, ISBN 978-0-387-97495-8. This will walk through all the classical Lie groups, introduce the connections with Lie algebras, and begin to teach representation theory (roughly, finding a group of matrices that mimics the group in question). Groups are now used heavily in physics, and you should tell our colleagues in physics of your special interests; they can better advise you of what they found helpful for their purposes (which are not the same as ours). --KSmrq^T 12:32, 15 July 2007 (UTC)

Thanks for the recommendations! I just ordered 5 books, most of which were recommended above, some combination of which will hopefully get me to where I want to be. MrRedact 04:39, 16 July 2007 (UTC)