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.

References

• Helgason, Sigurdur (2001). Differential Geometry, Lie Groups, and Symmetric Spaces. American Mathematical Society. ISBN 978-0-8218-2848-9. https://books.google.com/books?id=a9KFAwAAQBAJ.