Orthogonal symmetric Lie algebra

In mathematics, an orthogonal symmetric Lie algebra is a pair $(𝔤,s)$ consisting of a real Lie algebra $𝔤$ and an automorphism $s$ of $𝔤$ of order $2$ such that the eigenspace $𝔲$ of s corresponding to 1 (i.e., the set $𝔲$ 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 $𝔲$ intersects the center of $𝔤$ 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 $(𝔤,s)$ be effective orthogonal symmetric Lie algebra, and let $𝔭$ denotes the -1 eigenspace of $s$. We say that $(𝔤,s)$ is of compact type if $𝔤$ is compact and semisimple. If instead it is noncompact, semisimple, and if $𝔤=𝔲+𝔭$ is a Cartan decomposition, then $(𝔤,s)$ is of noncompact type. If $𝔭$ is an Abelian ideal of $𝔤$, then $(𝔤,s)$ is said to be of Euclidean type.

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

References

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