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Mark R. Sepanski

Compact Lie Groups (Graduate Texts in Mathematics, 235) Softcover reprint of hardcover 1st ed. 2007 Edition

by Mark R. Sepanski (Author)

4.5 13 ratings

Part of: Graduate Texts in Mathematics (179 books)

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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.

ISBN-10

1441921389
ISBN-13

978-1441921383
Edition

Softcover reprint of hardcover 1st ed. 2007
Publisher

Springer
Publication date

November 23, 2010
Part of series

Graduate Texts in Mathematics
Language

English
Dimensions

6.1 x 0.48 x 9.25 inches
Print length

214 pages
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Editorial Reviews

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"Groups serve to parameterize the symmetries of mathematical objects and the ways symmetries combine. … offers students willing to take a few things on faith a vital vista on a subject they may wish to pursue at the graduate level. Summing Up: Recommended. Upper-division undergraduates through professionals." (D. V. Feldman, CHOICE, Vol. v4 (3), November, 2007)

"The representation theory of compact groups is an old and venerable subject. The problems and solutions are now well understood and serve as a guide for the more advanced parts of the representation theory. … The present one is intended as a textbook within the reach of a good undergraduate student. … The reading should be pleasant both for students and for teachers preparing a course on the subject." (David A. Renard, Mathematical Reviews, Issue 2008 a)

“This book offers an introduction to the theories of compact Lie groups and of Lie algebras, which is organized in an unusual way. … For the ambitious reader many exercises are provided.” (A. Cap, Monatshefte für Mathematik, Vol. 158 (2), October, 2009)

From the Back Cover

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. Included is 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 necessary Lie algebra theory is also developed in the text with a streamlined approach focusing on linear Lie groups.

Key Features:

• Provides an approach that minimizes advanced prerequisites

• Self-contained and systematic exposition requiring no previous exposure to Lie theory

• Advances quickly to the Peter–Weyl Theorem and its corresponding Fourier theory

• Streamlined Lie algebra discussion reduces the differential geometry prerequisite and allows a more rapid transition to the classification and construction of representations

• Exercises sprinkled throughout

This beginning graduate-level text, aimed primarily at Lie Groups courses and related topics, assumes familiarity with elementary concepts from group theory, analysis, and manifold theory. Students, research mathematicians, and physicists interested in Lie theory will find this text very useful.

Product details

Publisher ‏ : ‎ Springer; Softcover reprint of hardcover 1st ed. 2007 edition (November 23, 2010)
Language ‏ : ‎ English
Paperback ‏ : ‎ 214 pages
ISBN-10 ‏ : ‎ 1441921389
ISBN-13 ‏ : ‎ 978-1441921383
Item Weight ‏ : ‎ 1 pounds
Dimensions ‏ : ‎ 6.1 x 0.48 x 9.25 inches

Best Sellers Rank: #4,738,260 in Books (See Top 100 in Books)
- #611 in Group Theory (Books)
- #801 in Abstract Algebra (Books)
- #1,155 in Linear Algebra (Books)

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4.5 13 ratings

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lim_bus

An outstanding introduction to representation theory

Reviewed in the United States 🇺🇸 on June 27, 2010

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 Introduction to Compact Lie Groups (Series in Pure Mathematics) , or as in Hochschild's The structure of Lie groups (Holden-Day series in mathematics) ), either A. Weil's proof, based on the Lefschetz fixed point formula, (as in Bump's Lie Groups (Graduate Texts in Mathematics) , or as in Wallach's Harmonic analysis on homogeneous spaces (Pure and applied mathematics, 19) ). The arguments in Simon's work Representations of Finite and Compact Groups (Graduate Studies in Mathematics ; V. 10) 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 Lie Groups; Lie Algebras , 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 Lie Groups: Beyond an Introduction , or as in Brocker-tom Dieck's Representations of Compact Lie Groups . 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 Lie Groups (Universitext) , which contains Borel-Weil construction too; it's scope is wider than Sepanski's, but it relies heavily on topological group actions, orbits, etc.

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genesis

an excellent book, tightly argued showing the elegance of the ...

Reviewed in the United Kingdom 🇬🇧 on October 10, 2017

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

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