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**Introduction à la théorie des groupes de Lie classiques. (Introduction to the theory of classical Lie groups).**
*(French)*
Zbl 0598.22001

Collection Méthodes. Paris: Hermann. 345 p. (1986).

Among all possible ways to introduce graduate students to Lie group theory, this book has chosen to start from a detailed study of (non- trivial) examples. The structure of classical matrix groups and their relations to geometry are studied by linear algebra tools first. Little by little, general definitions and methods are introduced, but only when a significant example is at hand and new notions can become truly helpful. Having turned down unnecessary generality and abstraction, the authors have achieved a very efficient introduction to a good understanding of Lie group structure.

The book is divided into seven chapters: structure and topology of the linear group (chapters 1 and 2), exponential mapping (chap. 3), orthogonal groups (chap. 4), unitary groups (chap. 5), symplectic groups (chap. 6), integration on manifolds and harmonic polynomials (chap. 7).

Throughout the book, the base field is \({\mathbb{R}}\) or \({\mathbb{C}}\). Turning pages over, the reader will find polar, Iwasawa, Cartan, and Bruhat decompositions for matrices, the Cartan-\(Von Neumann\) theorem on closed subgroups of GL(n), the differential of the exponential mapping, the basic facts about Clifford algebras, spinor groups, Pfaffians, the first homotopy group of classical groups, the geometric interpretation of the symmetric space GL(n,\({\mathbb{C}})/GL(n,{\mathbb{R}})\) by real forms of \({\mathbb{C}}^ n\), of Sp(n)/U(n) by Lagrangians of \({\mathbb{C}}^ 2\), the Hopf fibration, the Berger fibration, etc. Particularly attractive are paragraphs 4.5, 4.9 and 6.8 on low dimensional groups SO(3), SO(4), Sp(2)... and their isomorphisms.

Two chapters deserve a special mention. The 80 pages chapter 4 on orthogonal groups is by far the biggest of the book, and plays a central role. Besides O(p,q), O(n) and SO(n), it contains an introduction to homotopy, fibrations, coverings, Clifford algebras, differentiable structure of homogeneous spaces, Lie subalgebras and subgroups, and a proof of the Campbell-Hausdorff formula. This may look disordered at first sight; actually, it is quite coherent with the authors’ philosophy, of introducing general tools only when they can be easily accepted, and helpful. Similarly, the word ”submanifold” is defined on page 68 and ”manifold” on page 147 only, ”symmetric space” in an appendix to chapter 6, etc.

Chapter 7 is somewhat different from the rest: it is the only chapter on analysis, if one excepts a stealthy appearance of Haar measures in chapter 3. This last chapter deals with orientation, Stokes formula, Riemannian measure; it gives an insight into representation theory, for the case of SO(n) on the space \(L^ 2(S^{n-1})\), with decomposition by means of harmonic polynomials.

The book ends with a detailed alphabetical index and list of notations; it is enriched by 168 substantial exercises, and 4 problems. In spite of a slightly confusing double numbering of paragraphs (with or without parentheses), and a sometimes hesitating typography (Roman versus italics...), this book is very accessible; it should be recommended as a guide to graduate students who wish to understand matrix groups, and learn the bases of Lie group theory - and to any teacher who, as the reviewer, agrees with the authors’ pedagogy.

The book is divided into seven chapters: structure and topology of the linear group (chapters 1 and 2), exponential mapping (chap. 3), orthogonal groups (chap. 4), unitary groups (chap. 5), symplectic groups (chap. 6), integration on manifolds and harmonic polynomials (chap. 7).

Throughout the book, the base field is \({\mathbb{R}}\) or \({\mathbb{C}}\). Turning pages over, the reader will find polar, Iwasawa, Cartan, and Bruhat decompositions for matrices, the Cartan-\(Von Neumann\) theorem on closed subgroups of GL(n), the differential of the exponential mapping, the basic facts about Clifford algebras, spinor groups, Pfaffians, the first homotopy group of classical groups, the geometric interpretation of the symmetric space GL(n,\({\mathbb{C}})/GL(n,{\mathbb{R}})\) by real forms of \({\mathbb{C}}^ n\), of Sp(n)/U(n) by Lagrangians of \({\mathbb{C}}^ 2\), the Hopf fibration, the Berger fibration, etc. Particularly attractive are paragraphs 4.5, 4.9 and 6.8 on low dimensional groups SO(3), SO(4), Sp(2)... and their isomorphisms.

Two chapters deserve a special mention. The 80 pages chapter 4 on orthogonal groups is by far the biggest of the book, and plays a central role. Besides O(p,q), O(n) and SO(n), it contains an introduction to homotopy, fibrations, coverings, Clifford algebras, differentiable structure of homogeneous spaces, Lie subalgebras and subgroups, and a proof of the Campbell-Hausdorff formula. This may look disordered at first sight; actually, it is quite coherent with the authors’ philosophy, of introducing general tools only when they can be easily accepted, and helpful. Similarly, the word ”submanifold” is defined on page 68 and ”manifold” on page 147 only, ”symmetric space” in an appendix to chapter 6, etc.

Chapter 7 is somewhat different from the rest: it is the only chapter on analysis, if one excepts a stealthy appearance of Haar measures in chapter 3. This last chapter deals with orientation, Stokes formula, Riemannian measure; it gives an insight into representation theory, for the case of SO(n) on the space \(L^ 2(S^{n-1})\), with decomposition by means of harmonic polynomials.

The book ends with a detailed alphabetical index and list of notations; it is enriched by 168 substantial exercises, and 4 problems. In spite of a slightly confusing double numbering of paragraphs (with or without parentheses), and a sometimes hesitating typography (Roman versus italics...), this book is very accessible; it should be recommended as a guide to graduate students who wish to understand matrix groups, and learn the bases of Lie group theory - and to any teacher who, as the reviewer, agrees with the authors’ pedagogy.

Reviewer: F.Rouvière

### MSC:

22-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to topological groups |

20G20 | Linear algebraic groups over the reals, the complexes, the quaternions |

22E15 | General properties and structure of real Lie groups |

57T20 | Homotopy groups of topological groups and homogeneous spaces |

20-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to group theory |