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**Compact Lie groups.**
*(English)*
Zbl 1246.22001

Graduate Texts in Mathematics 235. New York, NY: Springer (ISBN 978-0-387-30263-8/hbk; 978-1-4419-2138-3/ebook). xiv, 198 p. (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 representation theory. There are many excellent books discussing this theory. The present one is intended as a textbook within the reach of a good undergraduate student. Only Lie groups are considered. The basics of the theory are developed relatively briefly (200 pages), and only an elementary knowledge of differential geometry and functional analysis is assumed. Examples (classical groups) are carefully introduced. The reading should be pleasant both for students and for teachers preparing a course on the subject.

An overview of the contents follows: Chapter 1 starts with some basic notions (manifolds, Lie groups, homomorphisms, connectedness, simply connected covers, invariant measures). Then examples of classical Lie groups are given, including Spin groups. In Chapter 2, elementary finite-dimensional representation theory is developed: definitions and examples, operations on representations, unitarity, complete reducibility, the Schur lemma, canonical decomposition. Chapter 3 first discusses matrix coefficients and characters of finite-dimensional representations and then moves on to infinite-dimensional unitary representations. This requires some functional analysis (the spectral theorem for normal bounded operators for the Schur lemma, the spectral theorem for compact self-adjoint operators and Zorn’s lemma for the canonical decomposition, the Stone-Weierstrass theorem for the Peter-Weyl theorem). The regular representation, class functions, Fourier theory and projection operators are also discussed. Chapter 4 deals with Lie algebras of linear groups, exponential maps, Lie correspondence, covering homomorphisms. The structure theory of connected compact Lie groups is the subject of Chapter 5, the main result being the maximal torus theorem. It continues in Chapter 6 with root theory and associated structures (weights, \(\mathrm{sl}_2\)-triples, Weyl groups, Dynkin diagrams), and in Chapter 7 with highest weight theory, the Weyl integration formula, the Weyl character formula and finally the Borel-Weil theorem.

An overview of the contents follows: Chapter 1 starts with some basic notions (manifolds, Lie groups, homomorphisms, connectedness, simply connected covers, invariant measures). Then examples of classical Lie groups are given, including Spin groups. In Chapter 2, elementary finite-dimensional representation theory is developed: definitions and examples, operations on representations, unitarity, complete reducibility, the Schur lemma, canonical decomposition. Chapter 3 first discusses matrix coefficients and characters of finite-dimensional representations and then moves on to infinite-dimensional unitary representations. This requires some functional analysis (the spectral theorem for normal bounded operators for the Schur lemma, the spectral theorem for compact self-adjoint operators and Zorn’s lemma for the canonical decomposition, the Stone-Weierstrass theorem for the Peter-Weyl theorem). The regular representation, class functions, Fourier theory and projection operators are also discussed. Chapter 4 deals with Lie algebras of linear groups, exponential maps, Lie correspondence, covering homomorphisms. The structure theory of connected compact Lie groups is the subject of Chapter 5, the main result being the maximal torus theorem. It continues in Chapter 6 with root theory and associated structures (weights, \(\mathrm{sl}_2\)-triples, Weyl groups, Dynkin diagrams), and in Chapter 7 with highest weight theory, the Weyl integration formula, the Weyl character formula and finally the Borel-Weil theorem.

Reviewer: David Renard (Poitiers) (MR2279709)