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''<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" />''
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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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{{TEX|auto}}
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{{TEX|done}}
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036990/e0369906.png" /></td> </tr></table>
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'' $  S $
 +
with kernel  $  A $''
  
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
+
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
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036990/e03699017.png" /></td> </tr></table>
+
$$
 +
0  \rightarrow  A  \rightarrow  G  \mathop \rightarrow \limits ^  \phi    S  \rightarrow  0.
 +
$$
  
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.
+
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
  
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
+
$$
 +
[( s, a), ( s  ^  \prime  , a  ^  \prime  )]  = \
 +
([ s, s  ^  \prime  ], \alpha ( s) a  ^  \prime  - \alpha ( s  ^  \prime  ) a +
 +
[ a, a  ^  \prime  ]).
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036990/e03699031.png" /></td> </tr></table>
+
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.
  
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.
+
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|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====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  N. Jacobson,  "Lie algebras" , Interscience  (1962)  ((also: Dover, reprint, 1979))</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  N. Jacobson,  "Lie algebras" , Interscience  (1962)  ((also: Dover, reprint, 1979))</TD></TR></table>

Revision as of 19:38, 5 June 2020


$ 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))
How to Cite This Entry:
Extension of a Lie algebra. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Extension_of_a_Lie_algebra&oldid=46879
This article was adapted from an original article by A.K. Tolpygo (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article