Orthogonal symmetric Lie algebra

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

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

Every effective, orthogonal symmetric Lie algebra decomposes into a direct sum of ideals ${\displaystyle {\mathfrak {g}}_{0}}$, ${\displaystyle {\mathfrak {g}}_{-}}$ and ${\displaystyle {\mathfrak {g}}_{+}}$, each invariant under ${\displaystyle s}$ and orthogonal with respect to the Killing form of ${\displaystyle {\mathfrak {g}}}$, and such that if ${\displaystyle s_{0}}$, ${\displaystyle s_{-}}$ and ${\displaystyle s_{+}}$ denote the restriction of ${\displaystyle s}$ to ${\displaystyle {\mathfrak {g}}_{0}}$, ${\displaystyle {\mathfrak {g}}_{-}}$ and ${\displaystyle {\mathfrak {g}}_{+}}$, respectively, then ${\displaystyle ({\mathfrak {g}}_{0},s_{0})}$, ${\displaystyle ({\mathfrak {g}}_{-},s_{-})}$ and ${\displaystyle ({\mathfrak {g}}_{+},s_{+})}$ are effective orthogonal symmetric Lie algebras of Euclidean type, compact type and noncompact type.