Orthogonal Symmetric Lie Algebra
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In mathematics, an orthogonal symmetric Lie algebra is a pair (\mathfrak, s) 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 ...
\mathfrak and an automorphism s of \mathfrak of order 2 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 ...
\mathfrak of ''s'' corresponding to 1 (i.e., the set \mathfrak 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 \mathfrak intersects the center of \mathfrak 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, ...
, s being the differential of a symmetry. Let (\mathfrak, s) be effective orthogonal symmetric Lie algebra, and let \mathfrak denotes the -1 eigenspace of s. We say that (\mathfrak, s) is ''of compact type'' if \mathfrak 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 \mathfrak=\mathfrak+\mathfrak is a Cartan decomposition, then (\mathfrak, s) is ''of noncompact type''. If \mathfrak is an Abelian ideal of \mathfrak, then (\mathfrak, s) 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 \mathfrak_0, \mathfrak_- and \mathfrak_+, each invariant under s 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 \mathfrak, and such that if s_0, s_- and s_+ denote the restriction of s to \mathfrak_0, \mathfrak_- and \mathfrak_+, respectively, then (\mathfrak_0,s_0), (\mathfrak_-,s_-) and (\mathfrak_+,s_+) are effective orthogonal symmetric Lie algebras of Euclidean type, compact type and noncompact type.


References

* Lie algebras {{differential-geometry-stub