Abstract
Let (\( \mathfrak{g} \), τ) be a real simple symmetric Lie algebra and let W ⊂ \( \mathfrak{g} \) be an invariant closed convex cone which is pointed and generating with τ(W) = −W. For elements h ∈ \( \mathfrak{g} \) with τ(h) = h, we classify the Lie algebras \( \mathfrak{g} \)(W, τ, h) which are generated by the closed convex cones \( {C}_{\pm}\left(W,\tau, h\right):= \left(\pm W\right)\cap {\mathfrak{g}}_{\pm 1}^{-\tau }(h) \), where \( {\mathfrak{g}}_{\pm 1}^{-\tau }(h):= \left\{x\in \mathfrak{g}:\tau (x)=-x\left[h,x\right]=\pm x\right\} \). 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, τ, 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 τ of \( \mathfrak{g} \) with τ(W) = −W a list of possible subalgebras \( \mathfrak{g} \)(W, τ, h) up to isomorphy.
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OEH, D. CLASSIFICATION OF 3-GRADED CAUSAL SUBALGEBRAS OF REAL SIMPLE LIE ALGEBRAS. Transformation Groups 27, 1393–1430 (2022). https://doi.org/10.1007/s00031-020-09635-8
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DOI: https://doi.org/10.1007/s00031-020-09635-8