# Cartan decomposition

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