Orthogonal symmetric Lie algebra
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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 corrsponding 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.
Every orthogonal symmetric Lie algebra decomposes into a direct sum of ideals "of compact type", "of noncompact type" and "of Euclidean type".
- S. Helgason, Differential geometry, Lie groups, and symmetric spaces
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