Namespaces
Variants
Actions

Cartan decomposition

From Encyclopedia of Mathematics
Revision as of 17:21, 7 February 2011 by 127.0.0.1 (talk) (Importing text file)
(diff) ← Older revision | Latest revision (diff) | Newer revision → (diff)
Jump to: navigation, search

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)
How to Cite This Entry:
Cartan decomposition. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Cartan_decomposition&oldid=17328
This article was adapted from an original article by A.S. Fedenko (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article