Quasicentroid Of Lie Superalgebra

Derivation is a useful tool in the study of the structures, representations and classifications of Lie algebras and Lie superalgebras. Generalized derivation alge-bras are natural generalizations of derivation superalgebras concepts, its research is key point for Lie superalgebras structures and representations. The research for the centroid and quasicentroid is one of the most important subjects in the theory of Lie superalgebras. Centroid and quasicentroid is an important concept in the struc-tures of Lie superalgebras, centroid is an important subalgebra of generalized super derivation algebras, the elements of quasicentroid are special generalized derivation.The present thesis is devoted to studying over an algebraically closed field of characteristic inequal to 2, first we introduced the definition of generalized deriva-tion, centroid and quasicentroid in the Lie superalgebras and proved that generalized derivation algebras are subalgebras of generally linear Lie superalgebras, centroid is a subalgebra of generalized derivation superalgebras. We determine centroid of centerless Lie Superalgebras is Abel Lie superalgebras. By Fitting-decomposition , we concluded that quasicentroid as the multiplication of linear transformation is supercommutative, which is an Abel Lie superalgebra. Furthermore we studied properties of quasicentroids and proved the radical and nilradical keep invaraiant under the quasicentroid. By computing,we getted the images of even linear transfor-mation under Lie superalgebra in quasicentroid are in nilradical, thus we generalized the corresponding result of Lie superalgebras. We studed the relation between qua-sicentroid and centroid and proved the sufficient condition of quasicentroid equal to centroid, in particular, simple Lie superalgebras centroid equal to quasicentroid. In the end, we proved even linear transformation in the quasicentroid keep ideals invaraiant by the cohomology, furthermore even generalized derivation keep radical invaraiant.