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".

References

S. Helgason, Differential geometry, Lie groups, and symmetric spaces

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