Extensions of Lie superalgebras and their automorphisms

Abstract

The present dissertation contains four chapters and is devoted to the study of newlineinducibility of automorphisms for the abelian Lie superalgebra extensions and generalizing the results to the level of endomorphisms. Let 0 and#8594; a newlinei newlineand#8722;and#8594; e newlinep newlineand#8722;and#8594; g and#8594; 0 be an newlineabelian extension of a Lie superalgebra g by an abelian Lie superalgebra a. Every newlineautomorphism of e which keeps image of a invariant induces a pair of automorphisms newline(and#966;, and#968;) and#8712; Aut(a) × Aut(g). A famous inducibility problem asks when is a given pair newline(and#966;, and#968;) induced from an automorphism of e ? First we consider the problem in two newlineparticular cases, namely when can an automorphism of a be extended to an automorphism of e inducing identity map on g and when can an automorphism of g be lifted newlineto an automorphism of e fixing image of a pointwise? As a result we obtain two exact newlinesequences involving several automorphism groups and the second cohomology, which newlinein particular provide conditions for the above two questions. Next we combine, in a newlinesense, the above two exact sequences to construct a Wells-type exact sequence for Lie newlinesuperalgebras. This last exact sequence in particular gives conditions under which a newlinegiven pair (and#966;, and#968;) is inducible from an automorphism of e. We apply our above results to newlinethe case of various particular Lie superalgebras and obtain certain descriptions of automorphism groups of them. After this we try to generalise some of our results to the level newlineof endomorphisms and again obtain two exact sequences similar to the earlier two but newlinenow involving endomorphism groups replacing the earlier automorphism groups. The newlinefirst one is obtained as a consequence of the Hochschild-Serre spectral sequence while newlineto prove the second a direct approach has been sufficient. Among other corollaries of newlineour results we also establish two isomorphisms between certain automorphism groups newlineof semidirect product Lie superalgebras. newline