Elements in pointed invariant cones in Lie algebras and corresponding affine pairs (Q2119283)

This paper is concerned with the study of a finite dimensional Lie algebra \(\mathfrak{g}\) the set of all those elements \(x\) for which the closed convex hull of the adjoint orbit contains no affine lines. And the authors also study affine pairs \((x, h)\in \mathfrak{g}\times \mathfrak{g}\) for a pointed invariant cone \(W \subseteq \mathfrak{g}\). These pairs are characterized by the relations \(x \in W\) and \([h, x] = x\). The authors say: ``Convexity properties of adjoint orbits \(\mathcal{O}_{x} = \)Inn\((\mathfrak{g})x\) in a finite dimensional real Lie algebra, where Inn\((\mathfrak{g}) = \langle e^{ad\mathfrak{g}} \rangle\) is the group of inner automorphisms, play a role in many contexts. The interest in affine pairs stems from their relevance in Algebraic Quantum Field Theory (AQFT), where they arise from unitary representations \((U,\mathcal{H})\) of a corresponding Lie group \(G\) and their positive cones \(W = C_{U}\).'' Let \(\mathfrak{l}\) be a Lie algebra, \(V\) an \(\mathfrak{l}\)-module, \(\mathfrak{z}\) a vector space, and \(\beta : V \times V \to \mathfrak{z}\) an \(\mathfrak{l}\)-invariant skew-symmetric bilinear map. Then \(\mathfrak{z} \times V \times \mathfrak{l}\) is a Lie algebra with respect to the bracket: \([(z, v, x), (z^{'}, v^{'}, x^{'})] = (\beta(v, v^{'}), x.v^{'}-x^{'}.v, [x, x^{'}])\). We write \(\mathfrak{g}(\mathfrak{l}, V, \mathfrak{z}, \beta)\) for the so-obtained Lie algebra. The following definitions are quoted from this paper: \(\cdot\) For \(x \in \mathfrak{g}\), co(\(x\)) := \(\overline{\mathrm{conv}(\mathcal{O}_{x})}\) for the closed convex hull of \(\mathcal{O}_{x}\), \(C_{x} := \)cone(\(\mathcal{O}_{x}) = \overline{\mathbb{R}_{+}\mathrm{conv}(\mathcal{O}_{x})}\) for the closed convex cone generated by \(x\), \(\mathfrak{g}_{co} := \{x \in \mathfrak{g} : \)co\((x)\) pointed\} \(\supseteq \mathfrak{g}_{c} := \{x \in\mathfrak{g} : C_{x}\) pointed\}. \(\cdot U : G \to \)U\((\mathcal{H})\) is a unitary representation and \(\partial U(x)\) denotes the infinitesimal generator of the unitary one-parameter group \((U(\)exp\( tx))_{t\in \mathbb{R}}\), \(C_{U} := \{x \in \mathfrak{g}: -i\partial U(x) \ge 0\}\) for the positive cone of \(U\). \(\cdot\) For \(x = x_{\mathfrak{z}} + x_{V} + x_{\mathfrak{l}} \in \mathfrak{g}\), we consider the \(\mathfrak{z}\)-valued Hamiltonian function \(H^{\mathfrak{z}}_{x}: V \to \mathfrak{z}\), \(H^{\mathfrak{z}}_{x}(v) := P_{\mathfrak{z}}(e^{ad v}x) = x_{\mathfrak{z}} + [v, x_{V}] + \frac{1}{2}[v, [v, x_{\mathfrak{l}}]]\) \((P_{\mathfrak{z}}:\mathfrak{g} \to \mathfrak{z}\) projection), co\(_{\mathfrak{z}}(x) := \overline{\mathrm{conv}(H^{\mathfrak{z}}_{x} (V))}\). \(\cdot\) A finite dimensional real Lie algebra \(\mathfrak{g}\) is called \(admissible\) if it contains an Inn(\(\mathfrak{g}\))-invariant pointed generating closed convex subset \(C\). \(\cdot\) For \(\mathfrak{g} = \mathfrak{g}(\mathfrak{l}, V , \mathfrak{z}, \beta), D_{\mathrm{can}}(z, v, x) := (z,\frac{1}{2}v, 0)\) for the canonical derivation. \(\cdot\) \(D \in \)der\((\mathfrak{g})\) is called an Euler derivation if \(D\) is diagonalizable with Spec(\(D) \subseteq \{-1, 0, 1\}\). \bigskip The central result in Sect. 3 is the Characterization Theorem 3.20: Let \(\mathfrak{g} = \mathfrak{g}(\mathfrak{l}, V, \mathfrak{z}, \beta \)) be admissible such that the representation of \(\mathfrak{l}\) on \(V\) is faithful. Then, the convexity properties of the adjoint orbit \(\mathcal{O}_{x}\) are characterized as the properties of the Hamiltonian function \(H^{\mathfrak{z}}_{x}\), i.e., (a) co(\(x\)) is pointed if and only if co\(_{\mathfrak{z}}(x)\) is pointed, (b) \(C_{x}\) is pointed if and only if co\(_{\mathfrak{z}}(x)\) is pointed and, if \(x_{\mathfrak{l}}\) is nilpotent, then co\(_{\mathfrak{z}}(x)\) is contained in a pointed cone. The first main result on affine pairs is the Existence Theorem 4.7: Let \(\mathfrak{g} = \mathfrak{g}(\mathfrak{l}, V, \mathfrak{z}, \beta)\) be an admissible Lie algebra and \(x = x_{\mathfrak{z}} + x_{\mathfrak{l}} \in \mathfrak{z} + \mathfrak{l}\) be an ad-nilpotent element for which co\((x)\) is pointed. Then there exists a derivation \(D \in D_{\mathrm{can}} + ad\mathfrak{g}\) with \(Dx = x\) and Spec\((D) \subseteq \{0,\pm \frac{1}{2} ,\pm1\}\). Any invariant cone \(W\) generated by \(W_{\mathfrak{l}} := W \cap \mathfrak{l}\) and a central cone \(W_{\mathfrak{z}} \subseteq \mathfrak{z}\) satisfies \(e^{\mathbb{R}D}W = W\). The second main result characterizes the existence of Euler derivations with this property, i.e., where we even have Spec\((D) \subseteq \{0,\pm1\}\) (Theorem 4.17).