Orthogonal symmetric Lie algebra

(Redirected from 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.