##
**Linear algebraic groups.
2nd ed.**
*(English)*
Zbl 0927.20024

Progress in Mathematics (Boston, Mass.). 9. Boston, MA: Birkhäuser. x, 334 p. (1998).

This book is a completely new version of the first edition [Linear Algebraic Groups (Progress in Mathematics 9, Birkhäuser, Boston) (1981; Zbl 0453.14022)]. The aim of the old book was to present the theory of linear algebraic groups over an algebraically closed field. Reading that book, as well as A. Borel [Linear Algebraic Groups (Benjamin, New York) (1969; Zbl 0186.33201)] and J. E. Humphreys [Linear Algebraic Groups (Graduate Texts in Mathematics 21, Springer-Verlag, Berlin) (1975; Zbl 0325.20039)], many people entered the research field of linear algebraic groups.

The present book has a wider scope. Its aim is to treat the theory of linear algebraic groups over arbitrary fields. Again, the author keeps the treatment of prerequisites self-contained. The material of the first ten chapters covers the contents of the old book, but the arrangement is somewhat different and there are additions, such as the basic facts about algebraic varieties and algebraic groups over a ground field, as well as an elementary treatment of Tannaka’s theorem. These chapters can serve as a text for an introductory course on linear algebraic groups. The last seven chapters are new. They deal with algebraic groups over arbitrary fields. Some of the material has not been dealt with before in other texts, such as Rosenlicht’s results about solvable groups in Chapter 14, the theorem of Borel and Tits on the conjugacy over the ground field of maximal split tori in an arbitrary linear algebraic group in Chapter 15, and the Tits classification of simple groups over a ground field in Chapter 17. The book includes many exercises and a subject index.

The present book has a wider scope. Its aim is to treat the theory of linear algebraic groups over arbitrary fields. Again, the author keeps the treatment of prerequisites self-contained. The material of the first ten chapters covers the contents of the old book, but the arrangement is somewhat different and there are additions, such as the basic facts about algebraic varieties and algebraic groups over a ground field, as well as an elementary treatment of Tannaka’s theorem. These chapters can serve as a text for an introductory course on linear algebraic groups. The last seven chapters are new. They deal with algebraic groups over arbitrary fields. Some of the material has not been dealt with before in other texts, such as Rosenlicht’s results about solvable groups in Chapter 14, the theorem of Borel and Tits on the conjugacy over the ground field of maximal split tori in an arbitrary linear algebraic group in Chapter 15, and the Tits classification of simple groups over a ground field in Chapter 17. The book includes many exercises and a subject index.

Reviewer: Li Fuan (Beijing)

### MSC:

20G15 | Linear algebraic groups over arbitrary fields |

14L10 | Group varieties |

20-02 | Research exposition (monographs, survey articles) pertaining to group theory |

20Gxx | Linear algebraic groups and related topics |

14L17 | Affine algebraic groups, hyperalgebra constructions |

17B45 | Lie algebras of linear algebraic groups |

14-02 | Research exposition (monographs, survey articles) pertaining to algebraic geometry |

14L35 | Classical groups (algebro-geometric aspects) |

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

14Lxx | Algebraic groups |