Extension of a Lie algebra

$ S $ with kernel $ A $

A Lie algebra $ G $ with an epimorphism $ \phi : G \rightarrow S $ whose kernel is an ideal $ A \subset G $. This is equivalent to specifying an exact sequence

$$ 0 \rightarrow A \rightarrow G \mathop \rightarrow \limits ^ \phi S \rightarrow 0. $$

The extension is said to split if there is a subalgebra $ S _ {1} \subset G $ such that $ G = S _ {1} \oplus A $ (direct sum of modules). Then $ \phi $ induces an isomorphism $ S _ {1} \approx S $, and defines an action of the algebra $ S $ on $ A $ by derivations. Conversely, any homomorphism $ \alpha : S \rightarrow \mathop{\rm Der} A $, where $ \mathop{\rm Der} A $ is the algebra of derivations of $ A $, uniquely determines a split extension $ S \oplus A $ with multiplication given by

$$ [( s, a), ( s ^ \prime , a ^ \prime )] = \ ([ s, s ^ \prime ], \alpha ( s) a ^ \prime - \alpha ( s ^ \prime ) a + [ a, a ^ \prime ]). $$

For finite-dimensional Lie algebras over a field of characteristic 0, Lévy's theorem holds: If $ S $ is semi-simple, then every extension of $ S $ splits.

Of all non-split extensions, the Abelian ones have been studied most, i.e. the extensions with an Abelian kernel $ A $. In this case the action of $ G $ on $ A $ induces an action of $ G/A \cong S $ on $ A $, that is, $ A $ is an $ S $-module. For Lie algebras over a field, every Abelian extension of $ S $ with as kernel an $ S $-module $ A $ has the form $ S \oplus A $ with multiplication given by

$$ [( s, a), ( s ^ \prime , a ^ \prime )] = \ ([ s, s ^ \prime ], \alpha ( s) a ^ \prime - \alpha ( s ^ \prime ) a + \psi ( s, s ^ \prime )), $$

where $ \psi $ is some linear mapping $ S \wedge S \rightarrow A $. The Jacobi identity is equivalent to the fact that $ \psi $ is a two-dimensional cocycle (or $ 2 $-cocycle, see Cohomology of Lie algebras). The extensions determined by cohomologous cocycles are equivalent in a natural sense. In particular, an extension is split if and only if $ \psi $ is cohomologous to zero. Thus, the Abelian extensions of an algebra $ S $ with kernel $ A $ are described by the cohomology group $ H ^ {2} ( S, A) $. The study of extensions with solvable kernel reduces to the case of Abelian extensions.

References

[1]	N. Jacobson, "Lie algebras" , Interscience (1962) ((also: Dover, reprint, 1979))