The Weak C-ideals Of Lie Superalgebras And Their Related Properties

Ideals and subideals are the key research objects of the structure theory of Lie superalgebras.In this paper,the related problems of weak c-ideals of Lie algebras are extended to Lie superalgebras,and the weak c-ideals of Lie superalgebras and their related properties are studied.Then we study the relationship between solvable Lie superalgebras and weak c-ideals,and determine the range of the derived length of some solvable ideals.In the first part,the concept of weak c-ideals of Lie superalgebras is introduced on the basis field with characteristics not 2,3,and it is proved that the ideals of Lie superalgebras are c-complementary subalgebras and weak c-ideals.Then,using the Frattini ideals,we give a sufficient condition that a weak c-ideal of Lie superalgebras is its ideal;Then,the concept of weak c-simple Lie superalgebras is introduced,and it is proved that a simple Lie superalgebra if and only if it is a weak c-simple Lie superalgebra;Finally,the concept of weak c-ideals complementability is introduced,and the necessary and sufficient condition for a subalgebra of quotient algebra to have subideal complementation is proved.Then,using this necessary and sufficient condition,it is proved that a maximal subalgebra of Lie superalgebras is its weak c-ideal.In the second part,by using the weak c-ideals of Lie superalgebras on the base field with zero characteristic,we give the sufficient condition for the ideals of Lie superalgebras to be solvable.Then,by using some weak c-ideals of a class of Lie superalgebras,three sufficient conditions for the solvability of this class of Lie superalgebras are given;Then,the derived length ranges of some solvable ideals of Lie superalgebras on different basis fields are determined.