Book on Lie algebra & Lie groups for advanced math undergrad

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Scrumhalf
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Posting for my son (who does not have an account here):

He's a sophomore math major in college and is looking for a good book on Lie algebra and Lie Groups that he can study over the summer. He wants mathematical rigor, but he is thinking of grad school in theoretical physics, so he also wants the connection to physics.

He has an excellent math foundation: 1 full-year sequence on abstract algebra - groups, rings and fields (Dummit & Foote) and 1 full year sequence on real and complex analysis (baby Rudin, Wheeden and Zygmund, Ahlfors, Marshall). He will take a geometry/topology sequence next year, so he does not have formal training in that area yet, so the book cannot be topology-heavy.

He wants something that can build on his group theory and analysis knowledge and not spend too much time repeating stuff he already knows. Would Stillwell be the right level? Or Hall? Or would it be something that approaches the subject at a more advanced level?

Thanks!
 

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George Jones
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Posting for my son (who does not have an account here):

He's a sophomore math major in college and is looking for a good book on Lie algebra and Lie Groups that he can study over the summer. He wants mathematical rigor, but he is thinking of grad school in theoretical physics, so he also wants the connection to physics.

He has an excellent math foundation: 1 full-year sequence on abstract algebra - groups, rings and fields (Dummit & Foote) and 1 full year sequence on real and complex analysis (baby Rudin, Wheeden and Zygmund, Ahlfors, Marshall). He will take a geometry/topology sequence next year, so he does not have formal training in that area yet, so the book cannot be topology-heavy.

He wants something that can build on his group theory and analysis knowledge and not spend too much time repeating stuff he already knows. Would Stillwell be the right level? Or Hall? Or would it be something that approaches the subject at a more advanced level?
I am not familiar with Stillwell, which is aimed at undergraduates, but I do have a copy of the second edition of Hall (which I quite like), which is part of Springer's "Graduate Texts in Mathematics" series. Hall, however, is something that he can grow into. From the preface of Hall "The first four chapters of the book cover elementary Lie theory and could be used for an undergraduate course. ... Although I have tried to explain and motivate the results in Parts II and III of the book, using figures whenever possible, the material there is unquestionably more challenging than in Part I."

Also, Stillwell does not treat representations, which are the main applications of Lie algebras and groups in physics. From the beginning of Part II in Hall: "The results of this chapter are special cases of the general theory of representation theory of semisimple Lie algebras ... It is nevertheless useful to consider this case separately, in part because of the importance of SU(3) in physical applications, but mainly because seeing ... a simple example motivates the introduction of these structures later in a more general setting."
 
  • #4
martinbn
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If he doesn't have enough knowledge of differential geometry he might want to stick with Lie algebras only or find a book that has groups but is based on examples.
 
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Posting for my son (who does not have an account here):

He's a sophomore math major in college and is looking for a good book on Lie algebra and Lie Groups that he can study over the summer. He wants mathematical rigor, but he is thinking of grad school in theoretical physics, so he also wants the connection to physics.

He has an excellent math foundation: 1 full-year sequence on abstract algebra - groups, rings and fields (Dummit & Foote) and 1 full year sequence on real and complex analysis (baby Rudin, Wheeden and Zygmund, Ahlfors, Marshall). He will take a geometry/topology sequence next year, so he does not have formal training in that area yet, so the book cannot be topology-heavy.

He wants something that can build on his group theory and analysis knowledge and not spend too much time repeating stuff he already knows. Would Stillwell be the right level? Or Hall? Or would it be something that approaches the subject at a more advanced level?

Thanks!
I think this book could be a good stepping stone into theoretical physics.

It has the style that may not be to the liking of mathematicians but gives a good physical insight into the mathematics being used. There is a full chapter on the Lie Algebras and groups as used in the context of Particle physics and QFT.

For a more rigorous approach the GTM springer series book is highly recommended.
 
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I think this book could be a good stepping stone into theoretical physics.
For me it was terrible. Yes, I've learned some nice facts here and there, but Zee trying to be everyone-friendly makes it totally unfriendly. Besides, I think that almost total lack of rigour is not so good didactically. You learn a bunch of facts without even realising how much interrelated they are, because lack of rigour makes it impossible to show it properly.
 
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