Let ($$ \mathfrak{g} $$ 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.
Let $(G,\tau )$ be a finite-dimensional Lie group with an involutive automorphism $\tau $ of $G$ and let ${{\mathfrak{g}}} = {{\mathfrak{h}}} \oplus{{\mathfrak{q}}}$ be its corresponding Lie algebra decomposition. We show that every nondegenerate strongly continuous representation on a complex Hilbert space ${\mathcal{H}}$ of an open $^\ast $-subsemigroup $S \subset G$, where $s^{\ast } = \tau (s)^{-1}$, has an analytic extension to a strongly continuous unitary representation of the 1-connected Lie group $G_1^c$ with Lie algebra $[{{\mathfrak{q}}},{{\mathfrak{q}}}] \oplus i{{\mathfrak{q}}}$. We further examine the minimal conditions under which an analytic extension to the 1-connected Lie group $G^c$ with Lie algebra ${{\mathfrak{h}}} \oplus i{{\mathfrak{q}}}$ exists. This result generalizes the Lüscher–Mack theorem and the extensions of the Lüscher–Mack theorem for $^\ast $-subsemigroups satisfying $S = S(G^\tau )_0$ by Merigon, Neeb, and Ólafsson. Finally, we prove that nondegenerate strongly continuous representations of certain $^\ast $-subsemigroups $S$ can even be extended to representations of a generalized version of an Olshanski semigroup.
In this note, we study in a finite dimensional Lie algebra $${\mathfrak g}$$ g the set of all those elements x for which the closed convex hull of the adjoint orbit contains no affine lines; this contains in particular elements whose adjoint orbits generates a pointed convex cone $$C_x$$ C x . Assuming that $${\mathfrak g}$$ g is admissible, i.e., contains a generating invariant convex subset not containing affine lines, we obtain a natural characterization of such elements, also for non-reductive Lie algebras. Motivated by the concept of standard (Borchers) pairs in QFT, we also study pairs (x, h) of Lie algebra elements satisfying $$[h,x]=x$$ [ h , x ] = x for which $$C_x$$ C x pointed. Given x, we show that such elements h can be constructed in such a way that $$\mathop {\mathrm{ad}}\nolimits h$$ ad h defines a 5-grading, and characterize the cases where we even get a 3-grading.
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