In
mathematics, an orthogonal symmetric Lie algebra is a pair
consisting of a real
Lie algebra
In mathematics, a Lie algebra (pronounced ) is a vector space \mathfrak g together with an operation called the Lie bracket, an alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow \mathfrak g, that satisfies the Jacobi iden ...
and an
automorphism of
of order
such that the
eigenspace
In linear algebra, an eigenvector () or characteristic vector of a linear transformation is a nonzero vector that changes at most by a scalar factor when that linear transformation is applied to it. The corresponding eigenvalue, often denoted ...
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
In mathematics, a symmetric space is a Riemannian manifold (or more generally, a pseudo-Riemannian manifold) whose group of symmetries contains an inversion symmetry about every point. This can be studied with the tools of Riemannian geometry, ...
,
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
Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to:
* Interstate compact
* Blood compact, an ancient ritual of the Philippines
* Compact government, a type of colonial rule utilized in British ...
and
semisimple
In mathematics, semi-simplicity is a widespread concept in disciplines such as linear algebra, abstract algebra, representation theory, category theory, and algebraic geometry. A semi-simple object is one that can be decomposed into a sum of ''sim ...
. 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
The direct sum is an operation between structures in abstract algebra, a branch of mathematics. It is defined differently, but analogously, for different kinds of structures. To see how the direct sum is used in abstract algebra, consider a mo ...
of ideals
,
and
, each invariant under
and orthogonal with respect to the
Killing form
In mathematics, the Killing form, named after Wilhelm Killing, is a symmetric bilinear form that plays a basic role in the theories of Lie groups and Lie algebras. Cartan's criteria (criterion of solvability and criterion of semisimplicity) s ...
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.
References
*
Lie algebras
{{differential-geometry-stub