Abstract

We show that the classification of simple finite group schemes over an algebraically closed field reduces to the classification of abstract simple finite groups and of simple restricted Lie algebras in positive characteristic. Both these two simple objects have been classified. We review this classification. Finally, we address the problem of determining the infinitesimal deformations of simple finite group schemes.

Citations

386	Basic Algebra - Jacobson - 1974
303	Finite Groups of Lie Type - CARTER - 1985
264	Representations of Algebraic Groups - Jantzen - 1987
68	The Octonions - Baez
53	Farnsteiner: Modular Lie algebras and their representations. Monographs and Textbooks - Strade, R - 1988
52	Groupes algébriques. Tome I: Géométrie algébrique, généralités, groupes commutatifs, Avec un appendice Corps de classes local par Michiel Hazewinkel, Masson & Cie - Demazure, Gabriel - 1970
34	Simple Lie algebras over fields of positive characteristic, I. Structure theory, de Gruyter - Strade - 2004
20	Les groupes de transformations continus, infinis, simples - Cartan - 1909
16	Classes of restricted Lie algebras of characteristic p - Jacobson - 1941
15	Cohomology of restricted Lie algebras - Hochschild - 1954
14	Graded Lie algebras of finite characteristic - Kostrikin, Shafarevich - 1969
11	A brief history of the classification of the finite simple groups,Bull - Solomon
11	Classification of Simple Lie Algebras over Algebraically Closed Fields of Prime Characteristic,Bull - Strade, Wilson - 1991
10	The status of the classification of the finite simple groups - Aschbacher
8	Deformations of classical Lie algebras - Kuznetsov, Chebochko
8	Deformations of a Lie algebra of type G2 of characteristic three - Kirillov, Kuznetsov, et al.
8	On finite simple groups and their classification - Solomon - 1995
6	Cartan’s pseudogroups and the p-algebras of Lie (Russian),Dokl. Akad. Nauk SSSR,168 - Kostrikin, Shafarevich - 1966
5	Simple Lie algebras of characteristic p - Albert, Frank
5	Infinitesimal deformations of restricted simple Lie algebras II - Viviani
4	The restricted simple Lie algebras are of classical or Cartan type - Block, Wilson - 1984
4	Deformations of classical Lie algebras with a homogeneous root system in characteristic two - Chebochko
4	Cohomology groups of reduced enveloping algebras - Farnsteiner - 1991
4	The classification of the simple Lie algebras over a field with nonzero characteristic (Russian),Izv - Kac - 1970
4	Simple Lie algebras of characteristic 5 - Melikian - 1980
4	Deformations of simple Lie algebras - Rudakov - 1971
4	Restricted simple Lie algebras and their infinitesimal deformations, Proceedings Conference ”From Lie Algebras to Quantum Groups”, Centro Internacional de Matematica de Coimbra no. 28, 2006 (available at arXiv:math.RA/0702755 - Viviani
3	Trace forms on Lie algebras - Block - 1962
3	Two new classes of simple Lie algebras - Frank - 1964
3	Deformations of the restricted Melikian algebra - Viviani
1	A new class of simple Lie algebras,Proc - Frank - 1954
1	Abstract derivations and Lie algebras,Trans - Jacobson - 1937
1	Infinitesimal deformations of restricted simple Lie algebras I - Vivani
1	Deformations of simple finite group schemes. Preprint available at arXiv:0705.0821 - Viviani
1	Introduction to Affine Group Schemes - Watherhouse - 1979