Extension of a Lie algebra
with kernel
A Lie algebra with an epimorphism
whose kernel is an ideal
. This is equivalent to specifying an exact sequence
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The extension is said to split if there is a subalgebra such that
(direct sum of modules). Then
induces an isomorphism
, and defines an action of the algebra
on
by derivations. Conversely, any homomorphism
, where
is the algebra of derivations of
, uniquely determines a split extension
with multiplication given by
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For finite-dimensional Lie algebras over a field of characteristic 0, Lévy's theorem holds: If is semi-simple, then every extension of
splits.
Of all non-split extensions, the Abelian ones have been studied most, i.e. the extensions with an Abelian kernel . In this case the action of
on
induces an action of
on
, that is,
is an
-module. For Lie algebras over a field, every Abelian extension of
with as kernel an
-module
has the form
with multiplication given by
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where is some linear mapping
. The Jacobi identity is equivalent to the fact that
is a two-dimensional cocycle (or
-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
is cohomologous to zero. Thus, the Abelian extensions of an algebra
with kernel
are described by the cohomology group
. 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)) |
Extension of a Lie algebra. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Extension_of_a_Lie_algebra&oldid=13207