P-envelopes And Toral Ranks Of Lie Superalgebras

p-envelopes and toral ranks of Lie algebras are both independent and relational subjects in the theory of Lie algebras. In the theory of Lie algebras over fields of characteristic p, toral ranks has been studied very well. Though module Lie algebras and module Lie superalgebras have large differences, the even part of a Lie superalgebra is a Lie algebra, the odd part of a Lie superalgebra has nature module structure of the even part. So study of Lie superalgebras are relational to Lie algebras. This paper have four problems in all as follows:The first chapter is an exordium, introduces the domestic and abroad research situation of Lie superalgebras and the source and the main contents of this paper.The second chapter is basic concepts, introduces the concepts of restricted Lie superalgebras and correlative concepts of toral ranks of Lie superalgebras.In the third chapter, we study some properties of restricted Lie superalgebras on the base of known theories of restricted Lie algebras, in particular, we study some properties of p-envelopes of Lie superalgebras.In the fourth chapter, we study some properties of toral ranks of Lie superal-gebras, which are similar to some results and investigative methods of Lie algebras.The innovation in this paper is study to p-envelopes of Lie superalgebras by using for reference foregone methods and academic fruits, and combining character-istics of p-envelopes of Lie algebras itself; As well as aiming at the problems of Lie algebras toral ranks, this paper study fully in the field of unmentioned Lie superal-gebras toral ranks. The paper extends some results of Lie superalgebras by being similar to study methods of p-envelopes and toral ranks of Lie algebras.The results lay a foundation to study restricted theories of Lie superalgebras.