##
**Basic theory of algebraic groups and Lie algebras.**
*(English)*
Zbl 0589.20025

Graduate Texts in Mathematics, 75. New York-Heidelberg-Berlin: Springer-Verlag. VIII, 267 p. DM 78.00 (1981).

The principal concern of this book is to use the theory of algebraic groups for exhibiting basic algebra (i.e. various basic techniques from field theory, multilinear algebra, commutative ring theory, algebraic geometry and general algebraic representation theory of groups and Lie algebras) in action. Accordingly, the emphasis is made on developing the major general mathematical tools used for gaining control over algebraic groups, rather than on securing the final definitive results, such as the classification of simple groups and their irreducible representations. The exposition is entirely self-contained; no detailed knowledge beyond the usual standard material of the first one or two years graduate study in algebra is presupposed. As the author points out in the preface, he was encouraged to write this exposition chiefly by the appearance of J. Humphrey’s book ”Linear Algebraic Groups” (1975; Zbl 0325.20039), where the required algebraic geometry has been cut down to a manageable size. In fact, the algebraic-geometric developments given in the book under review have resulted from Humphrey’s treatment simply by adding proofs of the underlying facts from commutative algebra. Moreover, much of the general structure theory in arbitrary characteristic has been adapted from A. Borel’s lecture notes ”Linear Algebraic Groups” (1969; Zbl 0186.332), and Humphreys’ book.

The contents is as follows: Ch.1. Representative functions and Hopf algebras. Ch.2. Affine algebraic sets and groups. Ch.3. Derivations and Lie algebras. Ch.4. Lie algebras and algebraic subgroups. Ch. 5. Semisimplicity and unipotency. Ch.6. Solvable groups. Ch.7. Elementary Lie algebra theory. Ch.8. Structure theory in characteristic 0 (sect. 1 reduces the study of unipotent algebraic groups completely to the study of representation-theoretically nilpotent Lie algebras; sect. 2 establishes the result that the tensor product of semisimple group representations is semisimple; sect. 3 deals with the ”algebraic hull” of Lie subalgebras L of the Lie algebra of an algebraic group - it is proved that \(L=[L,L]\) implies that L coincides with its algebraic hull; sect. 4 is devoted to the semi-direct decomposition of an algebraic group). Ch.9. Algebraic varieties. Ch.10. Morphisms of varieties and dimension. Ch.11. Local theory (the contents of this chapter is the dimension theory of local rings and its application to the investigation of tangent spaces to varieties and local properties of morphisms). Ch.12. Coset varieties. Ch.13. Borel subgroups. Ch.14. Application of Galois cohomology (this chapter concerns the field of definition of an algebraic group; sect. 1 provides some technical preparations; sect. 2 is devoted to the extension of the basic semidirect product decomposition for solvable groups from the case of an algebraically closed field to that of a perfect field; sect. 3 contains the crucial cross-section result that is used in sect. 4 for showing that if G is an irreducible algebraic group over an algebraically closed field, and H is a unipotent irreducible algebraic subgroup of G such that G/H is an affine variety then, as a variety, G is the direct product of G/H and H). Ch.15. Algebraic automorphism groups (sect. 1 gives a general description of the appropriate algebraic group structures for the ”algebraic” subgroups of the group W(G) of all algebraic group automorphisms of an algebraic group G; sect. 2 deals with the passage from W(G) to W(H), where H is an algebraic subgroup of G, and with the passage to W(G/H) in the case where H is normal in G; sect. 3 and 4 contain the principal results which concern the possibility of endowing W(G) with the structure of an algebraic group in such a way that G becomes a strict W(G)-variety, in particular, the theorem is proved which characterizes G such that W(G) is an algebraic group). Ch.16. Universal enveloping algebra. Ch.17. Semisimple Lie algebras (this is devoted entirely to the classical representation theory of semisimple Lie algebras via the theory of weights; the principal goal is the basic result that, if L is a finite-dimensional semisimple Lie algebra over a field of characteristic 0, then the continuous dual of the universal enveloping algebra is finitely generated as an algebra). Ch.18. From Lie algebras to groups (sect. 1 establishes the adjoint criterion for a Lie algebra to be that of an algebraic group; this is refined in sect. 2, where it is shown that the isomorphism classes of Lie algebras satisfying the adjoint criterion are in bijective correspondence with the isomorphism classes of irreducible algebraic groups with unipotent centers; sect. 3 concerns the basic facts on group coverings and simply connected groups; sect. 4 and 5 provide the construction of a simply connected algebraic group with a given Lie algebra).

Each chapter ends with a few notes ranging from supplementary results, amplifications of proofs, examples and counter-examples through exercises to references.

The contents is as follows: Ch.1. Representative functions and Hopf algebras. Ch.2. Affine algebraic sets and groups. Ch.3. Derivations and Lie algebras. Ch.4. Lie algebras and algebraic subgroups. Ch. 5. Semisimplicity and unipotency. Ch.6. Solvable groups. Ch.7. Elementary Lie algebra theory. Ch.8. Structure theory in characteristic 0 (sect. 1 reduces the study of unipotent algebraic groups completely to the study of representation-theoretically nilpotent Lie algebras; sect. 2 establishes the result that the tensor product of semisimple group representations is semisimple; sect. 3 deals with the ”algebraic hull” of Lie subalgebras L of the Lie algebra of an algebraic group - it is proved that \(L=[L,L]\) implies that L coincides with its algebraic hull; sect. 4 is devoted to the semi-direct decomposition of an algebraic group). Ch.9. Algebraic varieties. Ch.10. Morphisms of varieties and dimension. Ch.11. Local theory (the contents of this chapter is the dimension theory of local rings and its application to the investigation of tangent spaces to varieties and local properties of morphisms). Ch.12. Coset varieties. Ch.13. Borel subgroups. Ch.14. Application of Galois cohomology (this chapter concerns the field of definition of an algebraic group; sect. 1 provides some technical preparations; sect. 2 is devoted to the extension of the basic semidirect product decomposition for solvable groups from the case of an algebraically closed field to that of a perfect field; sect. 3 contains the crucial cross-section result that is used in sect. 4 for showing that if G is an irreducible algebraic group over an algebraically closed field, and H is a unipotent irreducible algebraic subgroup of G such that G/H is an affine variety then, as a variety, G is the direct product of G/H and H). Ch.15. Algebraic automorphism groups (sect. 1 gives a general description of the appropriate algebraic group structures for the ”algebraic” subgroups of the group W(G) of all algebraic group automorphisms of an algebraic group G; sect. 2 deals with the passage from W(G) to W(H), where H is an algebraic subgroup of G, and with the passage to W(G/H) in the case where H is normal in G; sect. 3 and 4 contain the principal results which concern the possibility of endowing W(G) with the structure of an algebraic group in such a way that G becomes a strict W(G)-variety, in particular, the theorem is proved which characterizes G such that W(G) is an algebraic group). Ch.16. Universal enveloping algebra. Ch.17. Semisimple Lie algebras (this is devoted entirely to the classical representation theory of semisimple Lie algebras via the theory of weights; the principal goal is the basic result that, if L is a finite-dimensional semisimple Lie algebra over a field of characteristic 0, then the continuous dual of the universal enveloping algebra is finitely generated as an algebra). Ch.18. From Lie algebras to groups (sect. 1 establishes the adjoint criterion for a Lie algebra to be that of an algebraic group; this is refined in sect. 2, where it is shown that the isomorphism classes of Lie algebras satisfying the adjoint criterion are in bijective correspondence with the isomorphism classes of irreducible algebraic groups with unipotent centers; sect. 3 concerns the basic facts on group coverings and simply connected groups; sect. 4 and 5 provide the construction of a simply connected algebraic group with a given Lie algebra).

Each chapter ends with a few notes ranging from supplementary results, amplifications of proofs, examples and counter-examples through exercises to references.

Reviewer: V.L.Popov

### MSC:

20G15 | Linear algebraic groups over arbitrary fields |

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

17B45 | Lie algebras of linear algebraic groups |

14L10 | Group varieties |

20G05 | Representation theory for linear algebraic groups |

20G10 | Cohomology theory for linear algebraic groups |

17B35 | Universal enveloping (super)algebras |

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