##
**Representation theory. A first course.**
*(English)*
Zbl 0744.22001

Graduate Texts in Mathematics. 129. New York etc.: Springer-Verlag,. xv, 551 p., 144 ill. (1991).

The representation theory of groups, in its present widely developed generality and depth, is certainly one of the main achievements of the 20th-century mathematics. Its ubiquity in many branches of contemporary mathematics and theoretical physics furnishes evidence for the central significance that representation theory has gained over the recent decades. In particular, the representation theory of Lie groups and Lie algebras has become an indispensable tool in almost all mathematical and physical theories dealing with group actions, geometry, and symmetries.

According to its vast development and ubiquitous significance, there are quite many introductory and reference texts on the representation theory of Lie groups and Lie algebras, varying in their degree of abstractness and/or completeness. The present textbook differs from them in various regards. First of all, it is designed as a textbook for beginners in the theory of finite-dimensional representations. Thus it focuses on the rather elementary part of representation theory, which is – in a certain sense – the common ground for more general developments.

Second, as probably the essential feature of the authors’ approach, the text concentrates primarily on typical, important and instructive examples. The corresponding general theory is not put in the first place; instead it is sparingly developed after the discussion of various examples, and that mainly as a unifying language to comprehend phenomena already encountered in various concrete cases. The authors do not strive after the utmost generality and efficiency in imparting the theory, but they guide their work along the didactic point of view. They do not hesitate to provide several proofs for one and the same fact, from different viewpoints, if this helps the reader to grasp the philosophy and the interrelations.

Third, the principle that beginners can best learn by working concrete examples dictates the text thoroughly. There are hundreds of exercises, of different purposes, difficulties and degrees of abstractness. Many of them indicate further directions of the theory, omitted topics, and some classical standard results. The section “Hints, Answers, and References”, at the end of the book, contains some additional information about those exercises which are particularly marked.

The fourth special feature of the book is that it bears a close relation to H. Weyl’s classic “The Classical Groups” [Princeton Univ. Press (1939; Zbl 0020.20601)], probably more than other modern texts do. In fact, the authors declare explicitly that one goal of their book is to present many of H. Weyl’s (and his predecessors’) ideas and results in a form more accessible to the contemporary generation of readers. Thereby the text makes a convenient bridge between the old and the new literature on that topic.

Now, as for the contents of the book, the text is divided into four main parts and six appendices. Part I provides, mainly for didactical reasons, an account on the representation theory of finite groups, with emphasis on what is useful for Lie groups later on.

Part II gives an introduction to Lie groups, Lie algebras, and their finite-dimensional representations. A great deal of effort is spent on explaining things in the case of low-dimensional concrete examples. This leads to some initial classification theory for Lie algebras of low dimension and rank.

Part III forms the heart of the book: the finite-dimensional representations of the classical groups. For each series of classical Lie algebras an explicit construction for representations of highest given weight is performed, and the geometric meaning of the decompositions of the naturally occuring representations is comprehensively discussed as well. A special emphasis is laid on exhibiting the various relations among the representations of the classical Lie groups, which are caused by algebraic relations between the Lie algebras.

Part IV, entitled “Lie Theory”, is less example-oriented than the previous parts. It is designed to throw a bridge between the foregoing, rather concrete examples and the general theory. The approach is to interpret the phenomena encountered before in the framework of the abstract theory and modern terminology. This includes the classification of complex simple Lie algebras by Dynkin diagrams, exceptional Lie algebras, homogeneous spaces, Bruhat decompositions, the general Weyl character formula, and multiplicity formulae.

The six appendices at the end of the book give complete proofs of some facts from the general theory of Lie algebras and invariants, as they were used in the course of the text. Among the topics explained here are: symmetric functions and determinantal identities, tensor and exterior algebra, semisimple Lie algebras, Cartan subalgebras, the Weyl group, Ado’s and Levi’s theorems, and invariant theory for the classical groups.

Altogether, the present textbook is an excellent introduction to the representation theory of Lie groups and Lie algebras. The prerequisites are minimal, and the amount of information, motivation, inspiration, and guidance is enormous. The reader is skillfully led to a level from which he can study the general theory, together with the many topics which had to be omitted here, with profit and appreciation. With regard to its main features mentioned in the beginning, this textbook is an outstanding example of didactic mastery, and it serves the purpose of the series “Readings in Mathematics” in a perfect manner.

According to its vast development and ubiquitous significance, there are quite many introductory and reference texts on the representation theory of Lie groups and Lie algebras, varying in their degree of abstractness and/or completeness. The present textbook differs from them in various regards. First of all, it is designed as a textbook for beginners in the theory of finite-dimensional representations. Thus it focuses on the rather elementary part of representation theory, which is – in a certain sense – the common ground for more general developments.

Second, as probably the essential feature of the authors’ approach, the text concentrates primarily on typical, important and instructive examples. The corresponding general theory is not put in the first place; instead it is sparingly developed after the discussion of various examples, and that mainly as a unifying language to comprehend phenomena already encountered in various concrete cases. The authors do not strive after the utmost generality and efficiency in imparting the theory, but they guide their work along the didactic point of view. They do not hesitate to provide several proofs for one and the same fact, from different viewpoints, if this helps the reader to grasp the philosophy and the interrelations.

Third, the principle that beginners can best learn by working concrete examples dictates the text thoroughly. There are hundreds of exercises, of different purposes, difficulties and degrees of abstractness. Many of them indicate further directions of the theory, omitted topics, and some classical standard results. The section “Hints, Answers, and References”, at the end of the book, contains some additional information about those exercises which are particularly marked.

The fourth special feature of the book is that it bears a close relation to H. Weyl’s classic “The Classical Groups” [Princeton Univ. Press (1939; Zbl 0020.20601)], probably more than other modern texts do. In fact, the authors declare explicitly that one goal of their book is to present many of H. Weyl’s (and his predecessors’) ideas and results in a form more accessible to the contemporary generation of readers. Thereby the text makes a convenient bridge between the old and the new literature on that topic.

Now, as for the contents of the book, the text is divided into four main parts and six appendices. Part I provides, mainly for didactical reasons, an account on the representation theory of finite groups, with emphasis on what is useful for Lie groups later on.

Part II gives an introduction to Lie groups, Lie algebras, and their finite-dimensional representations. A great deal of effort is spent on explaining things in the case of low-dimensional concrete examples. This leads to some initial classification theory for Lie algebras of low dimension and rank.

Part III forms the heart of the book: the finite-dimensional representations of the classical groups. For each series of classical Lie algebras an explicit construction for representations of highest given weight is performed, and the geometric meaning of the decompositions of the naturally occuring representations is comprehensively discussed as well. A special emphasis is laid on exhibiting the various relations among the representations of the classical Lie groups, which are caused by algebraic relations between the Lie algebras.

Part IV, entitled “Lie Theory”, is less example-oriented than the previous parts. It is designed to throw a bridge between the foregoing, rather concrete examples and the general theory. The approach is to interpret the phenomena encountered before in the framework of the abstract theory and modern terminology. This includes the classification of complex simple Lie algebras by Dynkin diagrams, exceptional Lie algebras, homogeneous spaces, Bruhat decompositions, the general Weyl character formula, and multiplicity formulae.

The six appendices at the end of the book give complete proofs of some facts from the general theory of Lie algebras and invariants, as they were used in the course of the text. Among the topics explained here are: symmetric functions and determinantal identities, tensor and exterior algebra, semisimple Lie algebras, Cartan subalgebras, the Weyl group, Ado’s and Levi’s theorems, and invariant theory for the classical groups.

Altogether, the present textbook is an excellent introduction to the representation theory of Lie groups and Lie algebras. The prerequisites are minimal, and the amount of information, motivation, inspiration, and guidance is enormous. The reader is skillfully led to a level from which he can study the general theory, together with the many topics which had to be omitted here, with profit and appreciation. With regard to its main features mentioned in the beginning, this textbook is an outstanding example of didactic mastery, and it serves the purpose of the series “Readings in Mathematics” in a perfect manner.

Reviewer: Werner Kleinert (Berlin)

### MSC:

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

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

20G05 | Representation theory for linear algebraic groups |

22E46 | Semisimple Lie groups and their representations |

17B10 | Representations of Lie algebras and Lie superalgebras, algebraic theory (weights) |

22E60 | Lie algebras of Lie groups |

17B20 | Simple, semisimple, reductive (super)algebras |

17-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to nonassociative rings and algebras |