A Study Of Superalgebra With Bracket And Correlative Problem

In this thesis ,we construct a new superalgebral called superalgebra with bracket corresponding to Lie superalgebra, and study the trivial homology,the universal central extension and lifting of automorphisms and derivations of it.Chapter 1 is devoted to proof the classic properties of universal center extension ,such as the uniqueness of universal central extension, the sufficiency and necessary for the exisitence of universal central extension, the simplily split of universal central extension(i.e.,Ifπ:A→Bis the universal central extension of B,then A is simplily split ,that is A is isomorphic to its universal central extension i.e., Id : A→A is a universal central extension of A.) and so on. We use reversal method to show the general result: the sufficiency and necessary for the exisitence of universal central extension,which is also a revolutionary in my thesis.In chapter 2 we discuss the cohomology theory of superalgebra with bracket, construct the trivial homology of the algebra, and obtain a 5-term exact sequence corresponding to a given short exact sequence by this way, which implies the kernel of the universal central extension of superalgebra with bracket.We construct a universal central extension of superalgebra with bracket with the same kernel obtained in chapter 2, and study the lifting of automophisms and derivations of superalgebra with bracket in chapter 3.