On Isoclinism and Capability of Lie Superalgebras

Abstract

Many theorems and formulas of Lie superalgebras run quite parallel to Lie algebras, sometimes giving interesting results. So it is quite natural to extend the new concepts of Lie algebras immediately to Lie superalgebra case as the later type of algebras have wide applications in Physics, Mathematics and related theories. Isoclinism of Lie superalgebras and Schur multiplier has been defined and studied currently. The purpose of this thesis is to study more deeply the properties of isoclinism and the relation of Schur multiplier to capability. In this thesis it is shown that for finite dimensional Lie superalgebras of same dimension, the notion of isoclinism and isomorphism are equivalent. Furthermore, it is shown that covers of finite dimensional Lie superalgebras are isomorphic using the notion of isoclinism . For a Lie superalgebra L, the set of all superderivations of L whose image is contained in the center of L is known as central derivation of L and is denoted by SDerz(L). It is a subalgebra of superderivation algebra. This thesis presents the work on the central derivation of nilpotent Lie superalgebras which have nilindex 2. In particular, stem Lie superalgebras are characterized by their central derivations. Moreover, relation between SDerz(L) and stem Lie superalgebra is obtained for finite as well as infinite dimensional nilpotent Lie superalgebras with nilindex 2 and nonabelian finite dimensional nilpotent Lie superalgebra. In this thesis it is shown that distributive law holds for nonabelian tensor product of Lie superalgebras under certain direct sums. Thereby a rule for nonabelian exterior square of a Lie superalgebra is obtained. Capable Lie superalgebra is defined and then some characterization is given in this thesis. Specifically, it is proved that epicenter of a Lie superalgebra is equal to exterior square. All capable Lie superalgebras whose derived subalgebras have dimension at most one are classified. As an application to those results, it is shown that there exists at least one nonabelian nilpotent capable Lie superalgebra L of dimension a + b _ 3 where dim L = (a j b):