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Keywords:
Lie algebra; module; generalized semidirect sum
Lie algebra; module; generalized semidirect sum
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.
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