Isoclinism and Derivation of Lie Superalgebras and Lie Yamaguti Algebras

Abstract

This dissertation delves into various assertions related to the structure and dimension of the derivation superalgebra Sder(G) of a Lie superalgebra G and other derivation subsuperalgebras of Sder(G). A derivation subsuperalgebra SderI J (G) is considered where I and J are two ideals of a finitedimensional Lie superalgera G. The subsuperalgebra containing superderivations of G whose images are in I and that map J to zero is denoted as SderI J (G). Similarly, Sdern c (G) is defined such that it contains the superderivations δ of G that satisfy δ(g) ∈ [g, Gn] for all g ∈ G. Then the notion of nisoclinism between two Lie superalgebras is defined. It is shown that for finitedimensional Lie superalgebras G and H with an nisoclinism between them established by the pair (φ, θ) there exists an isomorphism from SderGn+1 Zn(G)(G) to SderHn+1 Zn(H)(H). Also, some necessary and sufficient conditions under which Sdern c (G) is isomorphic to certain special subsuperalgebras of Sder(G) are proved. As an equivalence relation, isoclinism plays an important role in the classification of all groups. This dissertation establishes the introduction of the concept of isoclinism to the relative central extensions of a pair of regular HomLie superalgebras and analyses some of their important generalizations. Firstly, the notion of isoclinism, which is an equivalence relation, is defined on the relative central extensions of a pair of regular HomLie superalgebras. Then the relation among the relative central extensions in an isoclinism family of a particular relative central extension is determined. In particular, the conditions required for the two notions, isoclinism and isomorphism to coincide is studied for the relative central extensions of a pair of regular HomLie superalgebras which are finite dimensional. In recent times, LieYamaguti algebras have been studied widely with greater interest. A study on Gderivations is carried out in this dissertation in association to LieYamaguti algebras. LieYamaguti algebras are one type of special algebras that satisfy the properties of a Lie algebra and a Lie triple system simultaneously. A Gderivation of a LieYamaguti algebra G is a derivation with respect to both the bilinear and trilinear operations where G is taken as an automorphism group of G. Some profound results having vital importance in the study of Gderivations are incorporated in this dissertation. Moreover, the relationship between Gderivations and other generalized derivations of LieYamaguti algebras is also established. In this dissertation, the notions of isoclinism and Schur multipliers are introduced for LieYamaguti algebras. Schur multiplier of a LieYamaguti algebra G is the second homology group H2(G, Z) of G with coefficients from C∗. The structural aspects of the covers of LieYamaguti algebras are determined when their Schur multipliers are of finite dimension. Further, as one of the main results, it is shown that the maximal stem extensions of LieYamaguti algebras are precisely same as their stem covers.