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**Lectures on Lie groups and Lie algebras.**
*(English)*
Zbl 0832.22001

London Mathematical Society Student Texts. 32. Cambridge: Cambridge Univ. Press. viii, 190 p., £29.95; $ 49.95/hbk (1995).

From the introduction: “This book consists of notes based on the three introductory courses given at the LMS-SERC Instructional Conference on Lie theory and algebraic groups held at Lancaster University in September 1993: Lie algebras by Roger Carter; Lie groups by Graeme Segal; algebraic groups by Ian Macdonald.” The purpose of these notes is to provide a short clear and “fast-moving” introduction to the aforementioned three subjects which is accessible for postgraduate students. As a rule only some short proofs are explained in detail and the more complicated theorems are illustrated by discussing examples.

The first part on “Lie algebras and root systems” (45 pages) is divided into four sections. The first one deals with some basic notions concerning Lie algebras and their representations, the second section is concerned with simple Lie algebras over the field of complex numbers and their classification by Dynkin diagrams, the basic facts from the finite dimensional representation theory of these Lie algebras are explained in the third section, and the last section explains how one obtains the various series of finite simple groups of Lie type from the corresponding Lie algebras.

The second part on “Lie Groups” (87 pages) starts with examples of matrix groups and gives in the first five sections a brief introduction into Lie groups and their homogeneous spaces. The remaining 12 sections deal with representation theory reaching from compact groups and the Peter-Weyl theorem to complexifications, representations in tensor spaces and the Borel-Weil theorem for the unitary group. In the last three sections representations of non-compact groups are discussed and some of the specific features are illustrated for \(\text{Sl} (2,R)\) and the three dimensional Heisenberg group.

In the third part on “Algebraic groups” (53 pages) the reader is first introduced to the very basic notions of affine varieties, then to affine algebraic groups, and after a motivating interlude, projective varieties and tangent spaces are presented. In the second half these notions are applied to define the Lie algebra of an algebraic group, explain the basic facts on homogeneous spaces, Borel subgroups, maximal tori, and finally the circle closes with the classification of reductive groups by their root datum.

According to my opinion this collection of lecture notes provides a possibility for a postgraduate student to learn the basic ideas in the covered topics in a minimal amount of time. I would even recommend the first and third part for graduate students. The second part might be a bit dangerous for an unexperienced student because he would have to worry in many places about finding the appropriate assumptions for the stated theorems such as assumptions on closedness, connectedness, smoothness etc. which are not made explicit in the text but have to be made to make the statements correct. I also had the impression that many of the proofs in this part should better be called “sketches of proofs”. But these are minor points of criticism with respect to the intention of these lecture notes. Even for the specialists in the field they present a nice overview over a well chosen selection of some of the most important results and techniques in the wide field of Lie theory.

The first part on “Lie algebras and root systems” (45 pages) is divided into four sections. The first one deals with some basic notions concerning Lie algebras and their representations, the second section is concerned with simple Lie algebras over the field of complex numbers and their classification by Dynkin diagrams, the basic facts from the finite dimensional representation theory of these Lie algebras are explained in the third section, and the last section explains how one obtains the various series of finite simple groups of Lie type from the corresponding Lie algebras.

The second part on “Lie Groups” (87 pages) starts with examples of matrix groups and gives in the first five sections a brief introduction into Lie groups and their homogeneous spaces. The remaining 12 sections deal with representation theory reaching from compact groups and the Peter-Weyl theorem to complexifications, representations in tensor spaces and the Borel-Weil theorem for the unitary group. In the last three sections representations of non-compact groups are discussed and some of the specific features are illustrated for \(\text{Sl} (2,R)\) and the three dimensional Heisenberg group.

In the third part on “Algebraic groups” (53 pages) the reader is first introduced to the very basic notions of affine varieties, then to affine algebraic groups, and after a motivating interlude, projective varieties and tangent spaces are presented. In the second half these notions are applied to define the Lie algebra of an algebraic group, explain the basic facts on homogeneous spaces, Borel subgroups, maximal tori, and finally the circle closes with the classification of reductive groups by their root datum.

According to my opinion this collection of lecture notes provides a possibility for a postgraduate student to learn the basic ideas in the covered topics in a minimal amount of time. I would even recommend the first and third part for graduate students. The second part might be a bit dangerous for an unexperienced student because he would have to worry in many places about finding the appropriate assumptions for the stated theorems such as assumptions on closedness, connectedness, smoothness etc. which are not made explicit in the text but have to be made to make the statements correct. I also had the impression that many of the proofs in this part should better be called “sketches of proofs”. But these are minor points of criticism with respect to the intention of these lecture notes. Even for the specialists in the field they present a nice overview over a well chosen selection of some of the most important results and techniques in the wide field of Lie theory.

Reviewer: K.-H.Neeb (Erlangen)

### MSC:

22-02 | Research exposition (monographs, survey articles) pertaining to topological groups |

17-02 | Research exposition (monographs, survey articles) pertaining to nonassociative rings and algebras |

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

20G15 | Linear algebraic groups over arbitrary fields |

20D06 | Simple groups: alternating groups and groups of Lie type |