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 corresponding 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.

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

Every effective, orthogonal symmetric Lie algebra decomposes into a direct sum of ideals $$\mathfrak{g}_0$$, $$\mathfrak{g}_-$$ and $$\mathfrak{g}_+$$, each invariant under $$s$$ and orthogonal with respect to the Killing form of $$\mathfrak{g}$$, and such that if $$s_0$$, $$s_-$$ and $$s_+$$ denote the restriction of $$s$$ to $$\mathfrak{g}_0$$, $$\mathfrak{g}_-$$ and $$\mathfrak{g}_+$$, respectively, then $$(\mathfrak{g}_0,s_0)$$, $$(\mathfrak{g}_-,s_-)$$ and $$(\mathfrak{g}_+,s_+)$$ are effective orthogonal symmetric Lie algebras of Euclidean type, compact type and noncompact type.