Ideals,Derivations And LiesDer Pairs Of Lie Superalgebras

In this thesis,we aim to study ideals,derivations and LicsDer pairs of Lie supcralgcbras.It mainly contains ideals,c-ideals,quasi-ideals,derivations,centroids,cohomologies,deformations and central extensions.This thesis is divided into six chapters as follows:Chapter 1 is the introduction.This chapter introduces the research background and development status,as well as our works and the structure of this thesis.Chapter 2 is dedicated to three kinds of ideals of Lie superalgebras.Firstly,we define perfect ideals and the derived series of Lie superalgebras,and characterize the solvability of Lie superalgebras by using the maximal perfect ideal and the length of the derived series.Secondly,almost perfect ideals and the descending central series of Lie superalgebras are defined in this chapter.And by using the maximal near perfect ideal and the length of the descending central series,the nilpotency of Lie superalgebras is characterized.Finally,we define upper bound ideals and the upper central series of Lie superalgebras,and characterize the nilpotency of Lie superalgebras by using the minimal upper bound ideal and the length of the upper central series.Chapter 3 is concentrated on c-ideals and quasi-ideals of Lie superalgebras.Firstly,the c-ideal of Lie superalgebras is defined and the solvability and the supersolvability of Lie superalgebras are studied.Secondly,we classify Lie superalgebras whose all one-dimensional subalgebras are c-ideals.Finally,we define quasi-ideals,subquasi-ideals and core-free quasiideals of Lie superalgebras respectively,describe their relations and related properties,and classify core-free quasi-ideals of Lie superalgebras.Chapter 4 is concerned with super-biderivations of the contact modular Lie superalgebra.Firstly,we introduce the construction of the finite-dimensional contact modular Lie superalgebra K(m,n;t).Secondly,the weight space decomposition of K(m,n;t)on the canonical torus is given.Finally,the skew-symmetric super-biderivation on K(m,n;t)is described.By using the properties of the skew-symmetric super-biderivation on K(m,n;t),we construct a set of zero weight derivations on K(m,n;t)with skew-symmetric superbiderivations and canonical toral elements.And it is proved that the zero weight derivations acting on are inner derivations.Moreover,every skew-symmetric superbiderivation of K(m,n;t)is verified to be inner.Chapter 5 is committed to super-biderivations and linear super-commuting maps on current Lie superalgebras.Firstly,we define the current Lie superalgebra L?A,where L is a Lie superalgebra and A is an associative commutative algebra with unity.Secondly,we prove that if L is perfect and centerless,then every skew-symmetric super-biderivation on L?A is of the form of the centroid.Finally,if L is perfect and centerless,then each linear super-commuting map on L?A is in the centroid of L?A,which is demonstrated in this chapter.Chapter 6 is devoted to cohomologies,deformations and central extensions of a LiesDer pair.Firstly,we introduce the concept of a LiesDer pair.A representation of a LiesDer pair is defined and its corresponding cohomologies are studied.Secondly,formal deformations and deformations of order n of a LiesDer pair are investigated.We prove that the infinitesimals of two equivalent 1-parameter formal deformations of a LiesDer pair are in the same cohomology class.This chapter shows that a LiesDer pair is rigid if the second cohomology group is trivial,and a deformation of order n is extensible if and only if its obstruction class is trivial.We also study central extensions of a LiesDer pair.Central extensions of LiesDer pairs are classified by using the second cohomology group with the coefficient in the trivial representation.Finally,we prove that a pair of super-derivations is extensible if and only if its obstruction class is trivial.