Primitive Lie Superalgebras And Chief Factors Of Lie Superalgebras

The study of primitive Lie algebras and chief factors of Lie algebras has been very extensive.In recent years,many scholars applied some properties of chief factors to better study some structures of Lie algebras.In this paper,the problems related to primitive Lie algebras and the chief factors of Lie algebras are extended to Lie superalgebras and the related properties of primitive Lie superalgebras and the chief factors of Lie superalgebras are studied.In the first part,we introduce the concept of primitive Lie super algebras,that is,the Lie superalgebras with a core-free maximal subalgebra are called primitive Lie superalgebras;Then we study three types of primitive Lie superalgebras according to the property that there are at most two minimal ideals in a primitive Lie superalgebras,and the properties of three types of primitive Lie superalgebras are studied,we investigats that a Lie superalgebra L is primitive of type 1 or 3 if and only if there is a minimal ideal I of L complemented;Secondly,we give the necessary and sufficient conditions that a Lie superalgebra is the third type of primitive Lie superalgebra;Finally,we give the relation between simple Lie superalgebras and primitive Lie superalgebras of the second type.In the second part,we introduce the concept of chief factors of Lie superalgebras.According to the concept of the third type primitive Lie superalgebras,we give the L-connection relation between the chief factors of Lie superalgebras.Secondly,we introduce the concept and properties of supplemented chief factors.Finally,we introduce the concept of subalgebras that cover or avoid chief factors of Lie superalgebras,and further introduce the CAP-subalgebras of Lie superalgebras L,and prove that L is solvable if all maximal subalgebras of Lie superalgebra are CAP-subalgebras.