Orthogonal symmetric Lie algebra explained
In mathematics, an orthogonal symmetric Lie algebra is a pair
consisting of a real
Lie algebra
and an
automorphism
of
of order
such that the eigenspace
of
s corresponding to 1 (i.e., the set
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.
Let
be effective orthogonal symmetric Lie algebra, and let
denotes the -1 eigenspace of
. We say that
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. If
is an Abelian ideal of
, then
is said to be
of Euclidean type.
Every effective, orthogonal symmetric Lie algebra decomposes into a direct sum of ideals
,
and
, each invariant under
and orthogonal with respect to the
Killing form of
, and such that if
,
and
denote the restriction of
to
,
and
, respectively, then
,
and
are effective orthogonal symmetric Lie algebras of Euclidean type, compact type and noncompact type.
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