Classification of 3-graded causal subalgebras of real simple Lie algebras

Abstract

Let $(\mathfrak{g},\tau )$ be a real simple symmetric Lie algebra and let $W\subset \mathfrak{g}$ be an invariant closed convex cone which is pointed and generating with $\tau (W)=-W$. For elements $h\in \mathfrak{g}$ with $\tau (h)=h$, we classify the Lie algebras $\mathfrak{g}(W,\tau ,h)$ which are generated by the closed convex cones \[C_{\pm}(W,τ,h) := (\pm W) \cap \mathfrak{g}_{\pm 1}^{-τ}(h),\] where ${\mathfrak{g}}_{\pm 1}^{-\tau }(h):=\{x\in \mathfrak{g}:\tau (x)=-x,[h,x]=\pm x\}$. These cones occur naturally as the skew-symmetric parts of the Lie wedges of endomorphism semigroups of certain standard subspaces. We prove in particular that, if $\mathfrak{g}(W,\tau ,h)$ is non-trivial, then it is either a hermitian simple Lie algebra of tube type or a direct sum of two Lie algebras of this type. Moreover, we give for each hermitian simple Lie algebra and each equivalence class of involutive automorphisms $\tau$ of $\mathfrak{g}$ with $\tau (W)=-W$ a list of possible subalgebras $\mathfrak{g}(W,\tau ,h)$ up to isomorphy.

The title of the paper has been changed; the introduction has been rewritten; some of the proofs have been shortened

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