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Lie groups. 2nd ed. (English) Zbl 1279.22001

Graduate Texts in Mathematics 225. New York, NY: Springer (ISBN 978-1-4614-8023-5/hbk; 978-1-4614-8024-2/ebook). xiii, 551 p. (2013).
The book under review is the second edition. For the first edition (2004) [Zbl 1053.22001].
The book consist of four parts. The first part is entitled “Compact groups”. It contains such general facts as Schur orthogonality and the Peter-Weyl theorem.
The second part is entitled “Compact Lie groups”. At first vector fields on Lie groups and exponential mapping are discussed. Then the universal enveloping algebras are defined. After this the author describes representations of \(\mathfrak{sl}(2,\mathbb{C})\). Then, the universal cover and the local Frobenius theorem are explained and the author proceeds to the definition of root systems etc. For this purpose tori and maximum tori in Lie groups are discussed and a topological proof of Cartan’s theorem is given. After this, root systems are discussed, and examples are given (including noncrystallographic examples). Highest weights and the Weyl character formula are also explained in the second part. Finally, fundamental groups of Lie groups are discussed.
The third part is called “Noncompact Lie groups”. This part begins with a discussion of complexification, Coxeter groups, Borel subgroups and Bruhat decomposition. In particular Coxeter group representations of Weyl and affine Weyl groups are given, Demazure characters and the Bruhat order are also discussed. After this, symmetric spaces are investigated. An embedding of a noncompact symmetric space into its compact dual, boundary components and the Bergman-Shilov boundary of a symmetric tube domain and Cartan’s classification are discussed. Relative root systems are defined, embeddings of Lie groups are surveyed. Finally spin representations of orthogonal groups are explained.
The fourth part is called “Duality and other topics”. Following Howe, duality is a bijection between a set of representations of a group \(G\) and a set of representations of a group \(H\). There exists a representation \(\Omega\) of the group \(G\times H\), let \(\pi_i\otimes \pi'_i\) be an irreducible representation that occurs in the reducible representation \(\Omega\) and let \(\pi_i\otimes \pi'_i\) occur with multiplicity one. Then \(\pi\) is dual to \(\pi'\). Part four opens with a discussion of Mackey theory and the characters of \(\mathrm{GL}(n,\mathbb{C})\). Then, the Frobenius-Schur duality is proved. Applications of duality are given (the Jacobi-Trudy identity and computation of minors of Toeplitz matrices, traces of unitary matrices and random matrix theory). Also, \(\mathrm{GL}(n)\times\mathrm{GL}(m)\) duality is explained. The part ends with a discussion of Hecke algebras, the philosophy of cusp forms and the cohomology of Grassmanians.
If one compares the two editions one sees that a fair amount of new material is added. Some old material is rewritten. The new material includes the affine Weyl group, the Coxeter group, Demazure characters, Bruhat order, Schubert and Bott-Samelson varieties, the Borel-Weil theorem.
Also, the new edition contains an appendix on Sage – a computer algebra system dealing with Lie groups and algebras.
The book begins with a detailed explanation of the basic facts. The Parts 3 and 4 are less elementary. It contains a discussion of very nontrivial modern applications of Lie group theory in other areas of mathematics. These chapters contain a lot of material that cannot be found in other textbooks on Lie groups.
The text is very interesting and is superior to other textbooks on Lie group theory.

MSC:

22-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to topological groups
22Exx Lie groups
22C05 Compact groups
20Cxx Representation theory of groups
53C35 Differential geometry of symmetric spaces
11Fxx Discontinuous groups and automorphic forms

Citations:

Zbl 1053.22001

Software:

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