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