View source for Extension of a Lie algebra

''<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036990/e0369901.png" /> with kernel <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036990/e0369902.png" />''

A Lie algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036990/e0369903.png" /> with an epimorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036990/e0369904.png" /> whose kernel is an ideal <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036990/e0369905.png" />. This is equivalent to specifying an exact sequence

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The extension is said to split if there is a subalgebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036990/e0369907.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036990/e0369908.png" /> (direct sum of modules). Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036990/e0369909.png" /> induces an isomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036990/e03699010.png" />, and defines an action of the algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036990/e03699011.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036990/e03699012.png" /> by derivations. Conversely, any homomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036990/e03699013.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036990/e03699014.png" /> is the algebra of derivations of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036990/e03699015.png" />, uniquely determines a split extension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036990/e03699016.png" /> with multiplication given by

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For finite-dimensional Lie algebras over a field of characteristic 0, Lévy's theorem holds: If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036990/e03699018.png" /> is semi-simple, then every extension of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036990/e03699019.png" /> splits.

Of all non-split extensions, the Abelian ones have been studied most, i.e. the extensions with an Abelian kernel <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036990/e03699020.png" />. In this case the action of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036990/e03699021.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036990/e03699022.png" /> induces an action of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036990/e03699023.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036990/e03699024.png" />, that is, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036990/e03699025.png" /> is an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036990/e03699026.png" />-module. For Lie algebras over a field, every Abelian extension of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036990/e03699027.png" /> with as kernel an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036990/e03699028.png" />-module <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036990/e03699029.png" /> has the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036990/e03699030.png" /> with multiplication given by

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where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036990/e03699032.png" /> is some linear mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036990/e03699033.png" />. The Jacobi identity is equivalent to the fact that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036990/e03699034.png" /> is a two-dimensional cocycle (or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036990/e03699036.png" />-cocycle, see [[Cohomology of Lie algebras|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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036990/e03699037.png" /> is cohomologous to zero. Thus, the Abelian extensions of an algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036990/e03699038.png" /> with kernel <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036990/e03699039.png" /> are described by the cohomology group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036990/e03699040.png" />. The study of extensions with solvable kernel reduces to the case of Abelian extensions.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  N. Jacobson,   "Lie algebras" , Interscience  (1962)  ((also: Dover, reprint, 1979))</TD></TR></table>