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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