Orthogonal symmetric Lie algebra

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

Let $(\backslash mathfrak\{g\},\; s)$ be effective orthogonal symmetric Lie algebra, and let $\backslash mathfrak\{p\}$ denotes the -1 eigenspace of $s$. We say that $(\backslash mathfrak\{g\},\; s)$ is of compact type if $\backslash mathfrak\{g\}$ is compact and semisimple. If instead it is noncompact, semisimple, and if $\backslash mathfrak\{g\}=\backslash mathfrak\{u\}+\backslash mathfrak\{p\}$ is a Cartan decomposition, then $(\backslash mathfrak\{g\},\; s)$ is of noncompact type. If $\backslash mathfrak\{p\}$ is an Abelian ideal of $\backslash mathfrak\{g\}$, then $(\backslash mathfrak\{g\},\; s)$ is said to be of Euclidean type.

Every effective, orthogonal symmetric Lie algebra decomposes into a direct sum of ideals $\backslash mathfrak\{g\}\_0$, $\backslash mathfrak\{g\}\_-$ and $\backslash mathfrak\{g\}\_+$, each invariant under $s$ and orthogonal with respect to the Killing form of $\backslash mathfrak\{g\}$, and such that if $s\_0$, $s\_-$ and $s\_+$ denote the restriction of $s$ to $\backslash mathfrak\{g\}\_0$, $\backslash mathfrak\{g\}\_-$ and $\backslash mathfrak\{g\}\_+$, respectively, then $(\backslash mathfrak\{g\}\_0,s\_0)$, $(\backslash mathfrak\{g\}\_-,s\_-)$ and $(\backslash mathfrak\{g\}\_+,s\_+)$ are effective orthogonal symmetric Lie algebras of Euclidean type, compact type and noncompact type.