Difference between revisions of "Extension of a Lie algebra"
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| + | {{MSC|17B}} | ||
| − | + | '' $ 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|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==== | ||
| − | + | * {{Ref|1}} N. Jacobson, "Lie algebras" , Interscience (1962) {{ZBL|0121.27504}} (also: Dover, reprint, 1979) | |
Latest revision as of 12:53, 19 March 2023
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)
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=13207
Extension of a Lie algebra. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Extension_of_a_Lie_algebra&oldid=13207
This article was adapted from an original article by A.K. Tolpygo (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article