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

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, hright):= left(pm Wright)cap {mathfrak{g}}_{pm 1}^{-tau }(h) , where {mathfrak{g}}_{pm 1}^{-tau }(h):= left{xin mathfrak{g}:tau (x)=-xleft[h,xright]=pm xright} . 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.