Generalized semidirect sums of Lie algebras and their modules

Abstract

summary:Generalized semidirect sums of Lie algebras and their modules are introduced, which are not necessarily (non)-Abelian extensions and may be applied to construct Lie algebras from modules. Some properties of generalized semidirect sums are described. In particular, it is shown that finite dimensional non-solvable Lie algebras can be realized as generalized semidirect sums. The complete classification up to isomorphism of all generalized semidirect sums of $\mathfrak {sl}_2$ and its finite-dimensional irreducible modules is given