Lie algebras. (English) Zbl 0121.27504

Jacobson, Nathan

Interscience Tracts in Pure and Appl. Math. 10. New York-London: Interscience Publishers, a Division of John Wiley and Sons. ix, 331 p. (1962).

The theory of Lie algebras is developed with no reference to groups and to algebraic closedness of the ground field (if needed the latter is replaced by requiring a splitting Cartan subalgebra). The mathematical style is elementary and perspicuous. In details one can notice many improvements on prevailing methods, e.g. the exposition of the theorem on complete reducibility of linear representations of semisimple Lie algebras, Levi’s theorem, Ado’s theorem, and the direct construction of semisimple Lie algebras and their irreducible representations. Beside the classical theory and the subjects we mentioned before, the author deals with the character formulas, automorphisms of Lie algebras, and simple Lie algebras over arbitrary fields. The method of dealing with automorphisms is essentially analytical though formulated in terms of algebraic geometry. The results on automorphisms are of course used in the chapter on Lie algebras over arbitrary fields which reflects fairly well the state of our knowledge with respect to the classification of simple Lie algebras.

Reviewer: Hans Freudenthal (Utrecht)

MSC:

17Bxx	Lie algebras and Lie superalgebras
17-01	Introductory exposition (textbooks, tutorial papers, etc.) pertaining to nonassociative rings and algebras
17B05	Structure theory for Lie algebras and superalgebras
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

Keywords:

Lie algebras; representations of semisimple Lie algebras; Levi’s theorem; Ado’s theorem; character formulas; automorphisms of Lie algebras; simple Lie algebras over arbitrary fields