Orthogonal symmetric Lie algebra
In mathematics, an orthogonal symmetric Lie algebra is a pair [math]\displaystyle{ (\mathfrak{g}, s) }[/math] consisting of a real Lie algebra [math]\displaystyle{ \mathfrak{g} }[/math] and an automorphism [math]\displaystyle{ s }[/math] of [math]\displaystyle{ \mathfrak{g} }[/math] of order [math]\displaystyle{ 2 }[/math] such that the eigenspace [math]\displaystyle{ \mathfrak{u} }[/math] of s corresponding to 1 (i.e., the set [math]\displaystyle{ \mathfrak{u} }[/math] 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 [math]\displaystyle{ \mathfrak{u} }[/math] intersects the center of [math]\displaystyle{ \mathfrak{g} }[/math] 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, [math]\displaystyle{ s }[/math] being the differential of a symmetry.
Let [math]\displaystyle{ (\mathfrak{g}, s) }[/math] be effective orthogonal symmetric Lie algebra, and let [math]\displaystyle{ \mathfrak{p} }[/math] denotes the -1 eigenspace of [math]\displaystyle{ s }[/math]. We say that [math]\displaystyle{ (\mathfrak{g}, s) }[/math] is of compact type if [math]\displaystyle{ \mathfrak{g} }[/math] is compact and semisimple. If instead it is noncompact, semisimple, and if [math]\displaystyle{ \mathfrak{g}=\mathfrak{u}+\mathfrak{p} }[/math] is a Cartan decomposition, then [math]\displaystyle{ (\mathfrak{g}, s) }[/math] is of noncompact type. If [math]\displaystyle{ \mathfrak{p} }[/math] is an Abelian ideal of [math]\displaystyle{ \mathfrak{g} }[/math], then [math]\displaystyle{ (\mathfrak{g}, s) }[/math] is said to be of Euclidean type.
Every effective, orthogonal symmetric Lie algebra decomposes into a direct sum of ideals [math]\displaystyle{ \mathfrak{g}_0 }[/math], [math]\displaystyle{ \mathfrak{g}_- }[/math] and [math]\displaystyle{ \mathfrak{g}_+ }[/math], each invariant under [math]\displaystyle{ s }[/math] and orthogonal with respect to the Killing form of [math]\displaystyle{ \mathfrak{g} }[/math], and such that if [math]\displaystyle{ s_0 }[/math], [math]\displaystyle{ s_- }[/math] and [math]\displaystyle{ s_+ }[/math] denote the restriction of [math]\displaystyle{ s }[/math] to [math]\displaystyle{ \mathfrak{g}_0 }[/math], [math]\displaystyle{ \mathfrak{g}_- }[/math] and [math]\displaystyle{ \mathfrak{g}_+ }[/math], respectively, then [math]\displaystyle{ (\mathfrak{g}_0,s_0) }[/math], [math]\displaystyle{ (\mathfrak{g}_-,s_-) }[/math] and [math]\displaystyle{ (\mathfrak{g}_+,s_+) }[/math] are effective orthogonal symmetric Lie algebras of Euclidean type, compact type and noncompact type.
References
- Helgason, Sigurdur (2001). Differential Geometry, Lie Groups, and Symmetric Spaces. American Mathematical Society. ISBN 978-0-8218-2848-9. https://books.google.com/books?id=a9KFAwAAQBAJ.
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