In mathematics, an orthogonal symmetric Lie algebra is a pair
consisting of a real Lie algebra
and an automorphism
of order
such that the eigenspace
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
intersects the center of
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,
being the differential of a symmetry.
be effective orthogonal symmetric Lie algebra, and let
denotes the -1 eigenspace of
is of compact type if
is compact and semisimple.
If instead it is noncompact, semisimple, and if
is a Cartan decomposition, then
is of noncompact type.
is an Abelian ideal of
is said to be of Euclidean type.
Every effective, orthogonal symmetric Lie algebra decomposes into a direct sum of ideals
, each invariant under
and orthogonal with respect to the Killing form of
denote the restriction of
, respectively, then
are effective orthogonal symmetric Lie algebras of Euclidean type, compact type and noncompact type.
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