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.