Compact Lie Groups (Graduate Texts in Mathematics Book 235) 2007th Edition by Mark R. Sepanski (PDF)

Description

Blending algebra, analysis, and topology, the study of compact Lie groups is one of the most beautiful areas of mathematics and a key stepping stone to the theory of general Lie groups. Assuming no prior knowledge of Lie groups, this book covers the structure and representation theory of compact Lie groups. Coverage includes the construction of the Spin groups, Schur Orthogonality, the Peter-Weyl Theorem, the Plancherel Theorem, the Maximal Torus Theorem, the Commutator Theorem, the Weyl Integration and Character Formulas, the Highest Weight Classification, and the Borel-Weil Theorem. The book develops the necessary Lie algebra theory with a streamlined approach focusing on linear Lie groups.

User’s Reviews

Reviews from Amazon users which were colected at the time this book was published on the website:

⭐Sepanski’s book reaches its climax in chapter five, providing a clear and elementary proof of Cartan’s theorem on the conjugacy of maximal tori in a compact connected Lie group G (called here “Maximal Torus theorem”). The main tool is an old idea of G. A. Hunt (“A theorem of Elie Cartan”, Proc. A.M.S. 1956, a widely known work, not cited in his list of references). Of course, Sepanski adds new skills to complete his proof. It is remarkable that Hopf-Rinow theorem and the Lefschetz fixed point formula are not used. Moreover, in the same proof, he also deduces that the exponential map of G is onto. Outstanding! As far as I know, to achieve the same objective, most previous texts used either Hunt’s idea mixed with Hopf-Rinow theorem (as in Fegan’s

⭐, or as in Hochschild’s

⭐), either A. Weil’s proof, based on the Lefschetz fixed point formula, (as in Bump’s

⭐, or as in Wallach’s

⭐). The arguments in Simon’s work

⭐to prove Cartan’s conjugacy theorem seem similar to Sepanski’s, but years ago I was unable to understand them and I felt something was going wrong there (I then had to get back to Wallach’s book to finish the task). In Sepanski’s book, things are quite easy and crystal clear (though, somewhere I had to write some details). The rest of Sepanski’s work is fine, and includes Peter-Weyl theorem and its paraphernalia as well as Dynkin’s proof of the Baker-Campbell-Hausdorff formula (expX)(expY)=exp(X+Y+[X,Y]/2+…) (only for Lie subgroups of the general linear complex group GL(n,C)). The book assumes some previous knowledge of manifolds and invokes Fröbenius theorem to sketch the 1-1 correspondence beteween Lie connected subgroups H of a Lie group G with Lie algebra g, and the Lie subalgebras h of g (it could be deduced from Baker-Campbell-Hausdorff formula! see Hausner-Schwatz’s

⭐, pag 72). The rest of this work is more standard, treating roots and weights as simultaneous eigenvalues of representations of the algebra of a maximal torus (instead of thinking of them as simultaneous eigenvalues of representations of the torus itself, as in Knapp’s

⭐, or as in Brocker-tom Dieck’s

⭐. Of course, Sepanski’s book includes a detailed proof of H.Weyl’s character formula (here, the aforementioned Simon’s book is more comprehensive). It ends with Borel-Weil construction of all irreducible representations of compact semi-simple Lie groups. I think that the theory of representations of compact semi-simple Lie groups is much simpler and easy to understand than the (almost equivalent) theory of representations of complex semi-simple Lie algebras. For this task, Sepanski’s book is brief, direct and self-contained; therefore I fully recommend it. However, the cited books by Knapp and Bump add a rather extensive content on semi-simple real Lie algebras and their classification. On the other hand, Fegan’s little book is still valuable, but it should need a second edition to fill many details and gaps. I think that Sepanski’s book can also be compared with “Éléments d’analyse” vol. 5 by Jean Dieudonné, for its combination of simplicity, transparency and (relative) brevity. Finally I will cite Duistermaat-Kolk’s

⭐, which contains Borel-Weil construction too; it’s scope is wider than Sepanski’s, but it relies heavily on topological group actions, orbits, etc.

⭐an excellent book, tightly argued showing the elegance of the math but if you just want the basics its not for you

⭐Must by

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