‎N. Bourbaki ‎
‎Elements de mathematique Groupe et algebres de lie ‎

‎ Actualités scientifiques et industrielles , Hermann , Ray E2‎

Reference : 6087

‎"N.BOURBAKI,Elements de Mathematique XXVI. Croupes et Algebres de Lie: Chapitrel. Algebres de Lie (Hermann et cie), 148 pp.Chevalley completed two volumes of his well known work on Lie groups before he embarked on a systematic account of Lie algebras. In this latest volume of Bourbaki, which is the first in a series on Lie groups and algebras, we are first introduced to the theory of Lie algebras. A hundred pages of theory is balanced by over thirty pages of exercises. The theory in the case of a Lie algebra over a ring of prime characteristic is sketched in a number of the exercises. For the moment we confine our attention to those Lie algebras that are finite dimensional vector spaces over a field of charac- teristic zero.A number of criteria are obtained for a Lie algebra to be one of the following types: soluble, nilpotent, semi-simple. Varying soluble and nilpotent radicals play an important role. The soluble radical has a complement—a Levi subalgebra—which is semi-simple; two Levi subalgebras are transformed into each other by a special automorphism (Theorem of Levi-Mal'cev). The theory reaches its climax in the Theorem of Ado. Every Lie algebra has a faithful representation of finite dimension such that the image of the nilpotent radical consists entirely of nilpotent elements.The enveloping algebra of a Lie algebra over a commutative ring with unit element is considered in detail and the general form of the Poincare-Birkhoff-Witt Theorem on the isomorphism between the symmetric algebra and the graded algebra associated with the filtered enveloping algebra is obtained.The well known theory of Lazard on the connection between groups and Lie algebras is confined to a sketchy mention in the exercises and the following topics receive no attention: free Lie algebras, the Campbell-Hausdorff formula and basic monomials.The volume is almost self contained and can be recommended as an introduction to the theory of Lie algebras."Ray E2 ‎