Extension of a Lie algebra

2020 Mathematics Subject Classification: Primary: 17B [MSN][ZBL]

$S$ 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) Zbl 0121.27504 (also: Dover, reprint, 1979)