Cartan decomposition
A representation of a real non-compact semi-simple Lie algebra 
(cf. Lie algebra, semi-simple) as a direct sum of vector spaces (*). If 
denotes the complexification (complex envelope) of 
(cf. Complexification of a Lie algebra), then there exists in 
a real compact subalgebra 
of the same dimension as 
such that the following decompositions into direct sums of vector spaces hold:
 | (*) |
where 
is the subalgebra of invariant elements of some involutory automorphism (involution) 
of 
and 
is the set of anti-invariant elements of 
. The second formula is the Cartan decomposition of 
(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) |
Cartan decomposition. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Cartan_decomposition&oldid=46260