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Classification of simple Lie algebras. (Classification des algèbres de Lie simples.) (French) Zbl 0990.17008

Séminaire Bourbaki. Volume 1998/99. Exposés 850-864. Paris: Société Mathématique de France, Astérisque 266, 245-286, Exp. No. 858 (2000).
The paper under review is a survey on the state of the art (until the beginning of 1999) of an essential part of the structure theory of finite-dimensional simple Lie algebras over an algebraically closed ground field of prime characteristic \(p\). The introduction contains some brief remarks on the historical background and the connection of the classification problem for finite-dimensional modular simple Lie algebras to the corresponding problem for (in)finite-dimensional Lie algebras over an algebraically closed ground field of characteristic zero.
The author starts off by deriving Jacobson’s formula for the correction term in the additivity relation of the non-commutative \(p\)-th power on an associative algebra over \(\mathbb F_p\) from the integral Baker-Campbell-Hausdorff formula and by illustrating the notion of a restricted Lie algebra via several instructive examples. Then he introduces the modular analogues of the finite-dimensional complex simple Lie algebras, the so-called Lie algebras of classical type. This is followed by a detailed description of the restricted Lie algebras of Cartan type. Their derived series are determined and the simplicity of the last term in this series is proved. Instead of calculating with local coordinates the author uses de Rham cohomology. The classification result of R. E. Block and R. L. Wilson [J. Algebra 114, 115–259 (1988; Zbl 0644.17008)] is stated which says that every finite-dimensional restricted simple Lie algebra over an algebraically closed field of characteristic \(p>7\) is either of classical or of Cartan type. For \(p=7\) there seems to be no exception to their result. This slightly more general version of the theorem of Block-Wilson was conjectured by A. I. Kostrikin and I. R. Shafarevich in 1966. In characteristic \(p=5\) the 125-dimensional restricted Melikyan algebra is the only known restricted simple Lie algebra which is neither of classical nor of Cartan type (but closely related to the classical Lie algebra of type \(G_2\) and being a Gerstenhaber deformation of a Hamiltonian Lie algebra). At the time of the writing of this review a classification of the finite-dimensional restricted simple Lie algebras over an algebraically closed field of characteristic \(p>3\) seems to be within reach due to a recent paper of A. Premet and H. Strade [J. Algebra 242, 236–337 (2001; Zbl 0982.17010)]. For the remaining cases \(p=2\) and \(p=3\) there is not even a complete picture of what one can expect.
An intermediate section is devoted to indicate why the classification in the general case is much more complicated than the Block-Wilson classification. Contrary to the restricted case or the classification of finite-dimensional simple Lie algebras over an algebraically closed field of characteristic zero there exist continuous families of finite-dimensional modular simple Lie algebras (i.e., there are positive integers \(d\) with infinitely many isomorphism classes of \(d\)-dimensional modular simple Lie algebras). Moreover, the description of the relevant simple Lie algebras is much less explicit. In order to improve on the latter the author sketches a formalism which allows to obtain a description of Lie algebras of so-called generalized Cartan type by transforming the defining differential forms. Nevertheless, later on he gives the usual definition of the Lie algebras of generalized Cartan type by using an appropriate divided power algebra and its automorphisms. The classification result of H. Strade and R. L. Wilson [Bull. Am. Math. Soc. 24, 357–362 (1991; Zbl 0725.17023)] is stated which says that every finite-dimensional simple Lie algebra over an algebraically closed field of characteristic \(p>7\) is either of classical or of generalized Cartan type. As in the restricted case there seems to be no exception to this result for base fields of characteristic 7. In characteristic 5 the family of Melikyan algebras yields the only known simple Lie algebras which are neither of classical nor of generalized Cartan type (but again are closely related to the classical Lie algebra of type \(G_2\) and are Gerstenhaber deformations of Hamiltonian Lie algebras). Due to the nature of the Jacobi identity there are a lot of exceptional finite-dimensional simple Lie algebras over fields of characteristic 2 or 3. Finally, it should be mentioned that A. Premet and H. Strade in a series of papers [J. Algebra 189, 419–480 (1997; Zbl 0878.17019); J. Algebra 216, 190–301 (1999; Zbl 0939.17016); and the paper cited above] are working towards a classification of finite-dimensional simple Lie algebras over an algebraically closed field of characteristic \(p>3\).
The paper under review closes by some brief remarks about methods of proofs for the classification results, namely Cartan subalgebras and their adjoint actions on the corresponding modular simple Lie algebra resp. sandwich elements and their role in constructing an appropriate maximal subalgebra satisfying the recognition theorem due to V. G. Kac (as corrected by R. L. Wilson and refined by G. M. Benkart, T. B. Gregory, and A. Premet).
Altogether this very interesting paper deals not so much with the classification of modular simple Lie algebras (as the title might suggest) but with an attempt to give an explicit description of the Lie algebras of generalized Cartan type by using the calculus of the corresponding differential forms conceptually. Also a lot of quite detailed proofs are included which is not usual in an ordinary survey paper.
For the entire collection see [Zbl 0939.00019].

MSC:

17B50 Modular Lie (super)algebras
17-02 Research exposition (monographs, survey articles) pertaining to nonassociative rings and algebras
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