In mathematics, an orthogonal symmetric Lie algebra is a pair
(
g
,
s
)
consisting of a real Lie algebra
g
and an automorphism
s
of
g
of order
2
such that the eigenspace
u
of s corrsponding to 1 (i.e., the set
u
of fixed points) is a compact subalgebra. If "compactness" is omitted, it is called a symmetric Lie algebra. An orthogonal symmetric Lie algebra is said to be effective if
u
intersects the center of
g
trivially. In practice, effectiveness is often assumed; we do this in this article as well.
The canonical example is the Lie algebra of a symmetric space,
s
being the differential of a symmetry.
Every orthogonal symmetric Lie algebra decomposes into a direct sum of ideals "of compact type", "of noncompact type" and "of Euclidean type".
