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A representation of a real non-compact semi-simple Lie algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020510/c0205101.png" /> (cf. [[Lie algebra, semi-simple|Lie algebra, semi-simple]]) as a direct sum of vector spaces (*). If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020510/c0205102.png" /> denotes the complexification (complex envelope) of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020510/c0205103.png" /> (cf. [[Complexification of a Lie algebra|Complexification of a Lie algebra]]), then there exists in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020510/c0205104.png" /> a real compact subalgebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020510/c0205105.png" /> of the same dimension as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020510/c0205106.png" /> such that the following decompositions into direct sums of vector spaces hold:
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<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/c/c020/c020510/c0205107.png" /></td> <td valign="top" style="width:5%;text-align:right;">(*)</td></tr></table>
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where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020510/c0205108.png" /> is the subalgebra of invariant elements of some involutory automorphism (involution) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020510/c0205109.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020510/c02051010.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020510/c02051011.png" /> is the set of anti-invariant elements of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020510/c02051012.png" />. The second formula is the Cartan decomposition of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020510/c02051013.png" /> (see [[#References|[1]]]). The Cartan decomposition reduces the classification of real non-compact semi-simple Lie algebras to that of compact semi-simple Lie algebras and involutory automorphisms in them.
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A representation of a real non-compact semi-simple Lie algebra  $  \mathfrak g $(
 +
cf. [[Lie algebra, semi-simple|Lie algebra, semi-simple]]) as a direct sum of vector spaces (*). If  $  \mathfrak g ^ {\mathbf C } $
 +
denotes the complexification (complex envelope) of  $  \mathfrak g $(
 +
cf. [[Complexification of a Lie algebra|Complexification of a Lie algebra]]), then there exists in  $  \mathfrak g ^ {\mathbf C } $
 +
a real compact subalgebra  $  \mathfrak g  ^ {k} $
 +
of the same dimension as  $  \mathfrak g $
 +
such that the following decompositions into direct sums of vector spaces hold:
 +
 
 +
$$ \tag{* }
 +
\mathfrak g  ^ {k}  = \
 +
\mathfrak t + \mathfrak p ,\ \
 +
\mathfrak g  =  \mathfrak t +
 +
i \mathfrak p ,
 +
$$
 +
 
 +
where  $  \mathfrak t $
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is the subalgebra of invariant elements of some involutory automorphism (involution) $  \phi $
 +
of $  \mathfrak g  ^ {k} $
 +
and $  \mathfrak p $
 +
is the set of anti-invariant elements of $  \phi $.  
 +
The second formula is the Cartan decomposition of $  \mathfrak g $(
 +
see [[#References|[1]]]). The Cartan decomposition reduces the classification of real non-compact semi-simple Lie algebras to that of compact semi-simple Lie algebras and involutory automorphisms in them.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  S. Helgason,  "Differential geometry and symmetric spaces" , Acad. Press  (1962)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  S. Helgason,  "Differential geometry and symmetric spaces" , Acad. Press  (1962)</TD></TR></table>

Latest revision as of 10:08, 4 June 2020


A representation of a real non-compact semi-simple Lie algebra $ \mathfrak g $( cf. Lie algebra, semi-simple) as a direct sum of vector spaces (*). If $ \mathfrak g ^ {\mathbf C } $ denotes the complexification (complex envelope) of $ \mathfrak g $( cf. Complexification of a Lie algebra), then there exists in $ \mathfrak g ^ {\mathbf C } $ a real compact subalgebra $ \mathfrak g ^ {k} $ of the same dimension as $ \mathfrak g $ such that the following decompositions into direct sums of vector spaces hold:

$$ \tag{* } \mathfrak g ^ {k} = \ \mathfrak t + \mathfrak p ,\ \ \mathfrak g = \mathfrak t + i \mathfrak p , $$

where $ \mathfrak t $ is the subalgebra of invariant elements of some involutory automorphism (involution) $ \phi $ of $ \mathfrak g ^ {k} $ and $ \mathfrak p $ is the set of anti-invariant elements of $ \phi $. The second formula is the Cartan decomposition of $ \mathfrak g $( see [1]). The Cartan decomposition reduces the classification of real non-compact semi-simple Lie algebras to that of compact semi-simple Lie algebras and involutory automorphisms in them.

References

[1] S. Helgason, "Differential geometry and symmetric spaces" , Acad. Press (1962)
How to Cite This Entry:
Cartan decomposition. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Cartan_decomposition&oldid=46260
This article was adapted from an original article by A.S. Fedenko (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article