Superderivations, Cohomology and Isoclinism of Different Types of Lie Superalgebras

Abstract

This thesis investigates the structure and dimension of the superderivation algebra Der(L) of a Lie superalgebra L. First, a Lie superalgebra L is considered as a direct sum of two finite-dimensional Lie superalgebras G and H which have no non-trivial common direct factor and the dimension of Der(L) of L in terms of Der(G), Der(H), Hom(G,Z(H)) and Hom(H,Z(G)) is obtained. Furthermore, a Lie superalgebra L is taken as a semidirect sum of two Lie superalgebras G and H. Then the dimension of Der(L : H) is found, which is a subsuperalgebra of Der(L). This thesis introduces the notion of 3-Lie superalgebras equipped with a pair of superderivations. Initially, a representation of a 3-Lie superalgebra is considered, which establishes the first-order cohomologies by using a pair of superderivations of 3-Lie superalgebras. This induces a Lie superalgebra and its representation. Subsequently, an abelian extension of 3-Lie superalgebras of the form 0 ! A ! L ! B ! 0 with [A,A,L] = 0 is considered, which constructs an obstruction class to the extensibility of a compatible pair of superderivations of 3-Lie superalgebras. This thesis also studies the representation of multiplicative Hom-Lie supertriple systems and determines the low-dimensional cohomologies and coboundary operators of Hom Lie supertriple systems. The central extension theory for multiplicative Hom-Lie supertriple systems is introduced. Moreover, it confirms that one-to-one correspondence exists between equivalent classes of central extensions of a multiplicative Hom-Lie supertriple system with the third cohomology group. After that, the 1-parameter formal deformation of a multiplicative Hom-Lie supertriple system using cohomology is discussed. In recent years, the notion of isoclinism has been studied for Lie superalgebras. This thesis examines some properties of isoclinism for Hom-Lie superalgebras and n-Lie superalgebras. It is shown that isoclinism and isomorphism are equivalent for Hom-Lie superalgebras and n-Lie superalgebras.