Orthogonal symmetric Lie algebra

In mathematics, an orthogonal symmetric Lie algebra is a pair (\mathfrak{g}, s) consisting of a real Lie algebra \mathfrak{g} and an automorphism s of \mathfrak{g} of order 2 such that the eigenspace \mathfrak{u} of s corrsponding to 1 (i.e., the set \mathfrak{u} 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{u} intersects the center of \mathfrak{g} 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, s 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".

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