##
**Elements of mathematics. Lie groups and Lie algebras. Chapters 7 and 8.
Reprint of the 1975 original.
(Eléments de mathématique. Groupes et algèbres de Lie. Chapitres 7 et 8.)**
*(French)*
Zbl 1181.17001

Berlin: Springer (ISBN 3-540-33939-6). 271 p. (2006).

The volume under review is the faithful reprint of Chapters 7 and 8 of Book 9 within Nicolas Bourbaki’s fundamental and sweeping collection “Éléments de Mathématique”, following the French original edition published in 1975 by Hermann, Paris. Being part of a complete republishing program by Springer Verlag, which is to include all books and volumes of Bourbaki’s monumental work in their French original version and in affordable paperback form, the present volume nearly completes the reissue of Book 9 titled “Groupes et algèbres de Lie”. The original edition of Chapters 7 and 8 has been reviewed by Ian Stewart shortly after its appearance more than thirty years ago (Zbl 0329.17002), which we therefore may refer to in every detail.

However, as particularly this volume of Bourbaki’s entire treatise has become one of the most popular, most widely used, and most often quoted sources in the standard literature on Lie algebras, it seems appropriate and worthwile to recall the main contents and features of this important, rather specialized part of Book 9. Chapter 7 continues the study of Lie algebras by investigating the structure of their Cartan subalgebras, on the one hand, and regular elements of a Lie group on the other. The systematic and comprehensive treatment of these fundamental topics is carried out in five sections and two appendices, the contents of which are as follows:

Section 1 describes the weight space decomposition with respect to a family of linear representations of a Lie algebra, with special emphasis placed on nilpotent Lie algebras and semisimple Lie algebras. Section 2 is devoted to Cartan subalgebras and their construction by regular elements of Lie algebras, whereas Section 3 derives conjugacy theorems for Cartan subalgebras by using algebro-geometric arguments in terms of the Zariski topology. Section 4 introduces the concept of regular elements of a Lie group and discusses then their significance for the associated Lie algebra. Section 5 deals with splittable linear Lie algebras and their characterization, thereby enhancing earlier work of Mal’tsev and Jacobson. Appendix I briefly recalls the concept of Zariski topology in affine algebraic geometry, and Appendix II provides some topological and complex-analytic supplements regarding special connectedness properties.

Chapter 8 comprises thirteen sections, in which the main objects of study are semi-simple Lie algebras. Section 1 describes the representations of the Lie algebra \(\text{sl}(2,k)\) and their simple modules. Root systems of semi-simple Lie algebras are the basic objects studied in Section 2, while the following Section 3 is devoted to special subalgebras of semisimple Lie algebras such as stable subalgebras, Borel subalgebras, and parabolic subalgebras. Section 4 looks more closely at semisimple Lie algebras defined by reduced root systems, and Section 5 investigates the automorphisms of various classes of semisimple Lie algebras. The representation theory of semisimple Lie algebras is analyzed in the following sections, including simple modules of highest weight in Section 6, modules of finite dimension and their characters in Section 7, symmetric invariants in Section 8, and Weyl’s formula in Section 9. This is followed by a brief description of maximal subalgebras of semi-simple Lie algebras in Section 10, and then by the description of nilpotent elements, simple elements, principal elements, and \(\text{sl}_2\)-triples in Section 11. Lattices and Chevalley orders are explained in Section 12, before the classical Lie algebras of types \(A_1,B_1,C_1\) and \(D_1\) are presented in the concluding Section 13. Two tables illustrate the obtained results of this section as a summary, and the most important results on semisimple Lie algebras are once more compiled at the end of the current volume.

As usual and typical of Bourbaki’s books, each section comes with a wealth of complementing and further-leading exercises, for many of which detailed hints are given. No doubt, this volume was, is, and will remain one of the great source books in the general theory of Lie groups and Lie algebras.

However, as particularly this volume of Bourbaki’s entire treatise has become one of the most popular, most widely used, and most often quoted sources in the standard literature on Lie algebras, it seems appropriate and worthwile to recall the main contents and features of this important, rather specialized part of Book 9. Chapter 7 continues the study of Lie algebras by investigating the structure of their Cartan subalgebras, on the one hand, and regular elements of a Lie group on the other. The systematic and comprehensive treatment of these fundamental topics is carried out in five sections and two appendices, the contents of which are as follows:

Section 1 describes the weight space decomposition with respect to a family of linear representations of a Lie algebra, with special emphasis placed on nilpotent Lie algebras and semisimple Lie algebras. Section 2 is devoted to Cartan subalgebras and their construction by regular elements of Lie algebras, whereas Section 3 derives conjugacy theorems for Cartan subalgebras by using algebro-geometric arguments in terms of the Zariski topology. Section 4 introduces the concept of regular elements of a Lie group and discusses then their significance for the associated Lie algebra. Section 5 deals with splittable linear Lie algebras and their characterization, thereby enhancing earlier work of Mal’tsev and Jacobson. Appendix I briefly recalls the concept of Zariski topology in affine algebraic geometry, and Appendix II provides some topological and complex-analytic supplements regarding special connectedness properties.

Chapter 8 comprises thirteen sections, in which the main objects of study are semi-simple Lie algebras. Section 1 describes the representations of the Lie algebra \(\text{sl}(2,k)\) and their simple modules. Root systems of semi-simple Lie algebras are the basic objects studied in Section 2, while the following Section 3 is devoted to special subalgebras of semisimple Lie algebras such as stable subalgebras, Borel subalgebras, and parabolic subalgebras. Section 4 looks more closely at semisimple Lie algebras defined by reduced root systems, and Section 5 investigates the automorphisms of various classes of semisimple Lie algebras. The representation theory of semisimple Lie algebras is analyzed in the following sections, including simple modules of highest weight in Section 6, modules of finite dimension and their characters in Section 7, symmetric invariants in Section 8, and Weyl’s formula in Section 9. This is followed by a brief description of maximal subalgebras of semi-simple Lie algebras in Section 10, and then by the description of nilpotent elements, simple elements, principal elements, and \(\text{sl}_2\)-triples in Section 11. Lattices and Chevalley orders are explained in Section 12, before the classical Lie algebras of types \(A_1,B_1,C_1\) and \(D_1\) are presented in the concluding Section 13. Two tables illustrate the obtained results of this section as a summary, and the most important results on semisimple Lie algebras are once more compiled at the end of the current volume.

As usual and typical of Bourbaki’s books, each section comes with a wealth of complementing and further-leading exercises, for many of which detailed hints are given. No doubt, this volume was, is, and will remain one of the great source books in the general theory of Lie groups and Lie algebras.

Reviewer: Werner Kleinert (Berlin)

### MSC:

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

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

01A75 | Collected or selected works; reprintings or translations of classics |

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

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

17B40 | Automorphisms, derivations, other operators for Lie algebras and super algebras |

17B45 | Lie algebras of linear algebraic groups |