RESTRICTED LIE (SUPER)ALGEBRAS, CLASSIFICATION OF SIMPLE LIE SUPERALGEBRAS IN CHARACTERISTIC 2

Abstract

In characteristic 2, for Lie algebras, a (2,4)-structure is introduced in addition to the known, classical, restrictedness. For Lie superalgebras, the classical restrictedness of Lie algebras has two analogs: $(2|4)$- and $(2|2)$-structures; a $(2,4)|4$-structure on Lie superalgebras is the analog of a (2,4)-structure on the Lie algebras. In characteristic 2, two procedures that to every simple Lie algebra assign simple Lie superalgebras, most of them new, are offered. We proved that every simple finite-dimensional Lie superalgebra is obtained as the result of one of these procedures, so we described all simple finite-dimensional Lie superalgebras modulo non-existing at the moment classification of simple finite-dimensional Lie algebras. We give references to papers containing a conjectural method to obtain the latter classification and (currently incomplete) collections of examples of simple Lie algebras. In characteristic 3, we prove that the known exceptional simple finite-dimensional vectorial Lie (super)algebras are restricted.