CLASSIFICATION OF 3-GRADED CAUSAL SUBALGEBRAS OF REAL SIMPLE LIE ALGEBRAS

Abstract

AbstractLet ($$ \mathfrak{g} $$ g , τ) be a real simple symmetric Lie algebra and let W ⊂ $$ \mathfrak{g} $$ g be an invariant closed convex cone which is pointed and generating with τ(W) = −W. For elements h ∈ $$ \mathfrak{g} $$ g with τ(h) = h, we classify the Lie algebras $$ \mathfrak{g} $$ g (W, τ, h) which are generated by the closed convex cones $$ {C}_{\pm}\left(W,\tau, h\right):= \left(\pm W\right)\cap {\mathfrak{g}}_{\pm 1}^{-\tau }(h) $$ C ± W τ h ≔ ± W ∩ g ± 1 − τ h , where $$ {\mathfrak{g}}_{\pm 1}^{-\tau }(h):= \left\{x\in \mathfrak{g}:\tau (x)=-x\left[h,x\right]=\pm x\right\} $$ g ± 1 − τ h ≔ x ∈ g : τ x = − x h x = ± 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 $$ \mathfrak{g} $$ 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 $$ \mathfrak{g} $$ g with τ(W) = −W a list of possible subalgebras $$ \mathfrak{g} $$ g (W, τ, h) up to isomorphy.