arXiv subjects

Daniel Oeh

Publications and source records attributed to Daniel Oeh.

5 recordsLinked to original sources

arXiv2021

Elements in pointed invariant cones in Lie algebras and corresponding affine pairs

In this note we study in 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; this contains in particular elements whose adjoint orbits generates a pointed convex cone~$C_x$. Assuming that ${\mathfrak g}$ 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 $[h,x]=x$ for which $C_x$ pointed. Given $x$, we show that such elements $h$ can be constructed in such a way that ${\rm ad} h$ defines a $5$-grading, and characterize the cases where we even get a $3$-grading.

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arXiv2021

Nets of standard subspaces induced by antiunitary representations of admissible Lie groups I

Let $(\pi, \mathcal{H})$ be a strongly continuous unitary representation of a 1-connected Lie group $G$ such that the Lie algebra $\mathfrak{g}$ of $G$ is generated by the positive cone $C_\pi := \{x \in \mathfrak{g} : -i\partial \pi(x) \geq 0\}$ and an element $h$ for which the adjoint representation of $h$ induces a 3-grading of $\mathfrak{g}$. Moreover, suppose that $(\pi, \mathcal{H})$ extends to an antiunitary representation of the extended Lie group $G_\tau := G \rtimes \{\mathbf{1}, \tau_G\}$, where $\tau_G$ is an involutive automorphism of $G$ with $\mathbf{L}(\tau_G) = e^{i\pi\mathrm{ad} h}$. In a recent work by Neeb and \'Olafsson, a method for constructing nets of standard subspaces of $\mathcal{H}$ indexed by open regions of $G$ has been introduced and applied in the case where $G$ is semisimple. In this paper, we extend this construction to general Lie groups $G$, provided the above assumptions are satisfied and the center of the ideal $\mathfrak{g}_C = C_\pi - C_\pi$ of $\mathfrak{g}$ is one-dimensional. The case where the center of $\mathfrak{g}_C$ has more than one dimension will be discussed in a separate paper.

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arXiv2020

Lie wedges of endomorphism semigroups of standard subspaces in admissible Lie algebras

Let $\mathfrak{g}$ be a real finite-dimensional Lie algebra containing pointed generating invariant closed convex cones. We determine those derivations $D$ of $\mathfrak{g}$ which induce a 3-grading of the form $\mathfrak{g} = \mathfrak{g}_{-1} \oplus \mathfrak{g}_0 \oplus \mathfrak{g}_1$ on $\mathfrak{g}$ such that the $(\pm 1)$-eigenspaces $\mathfrak{g}_{\pm 1}$ of $D$ are generated by the intersections with generating cones of the form $W = \mathcal{O}_f^*$, where $\mathcal{O}_f$ is the coadjoint orbit of a linear functional $f \in \mathfrak{z}(\mathfrak{g})^*$ and $\mathcal{O}_f^*$ is the dual cone of $\mathcal{O}_f$. In particular, we show that, if $\mathfrak{g}$ is solvable, no such derivation except the trivial one exists. This continues our classification of Lie algebras generated by Lie wedges of endomorphism semigroups of standard subspaces. The classification is motivated by the relation of nets of standard subspaces to Haag-Kastler nets of von Neumann algebras in Algebraic Quantum Field Theory.

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arXiv2020

Classification of 3-graded causal subalgebras of real simple Lie algebras

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

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arXiv2018

Analytic extensions of representations of $*$-subsemigroups without polar decomposition

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 non-degenerate strongly continuous representation on a complex Hilbert space $\mathcal H$ of an open $*$-subsemigroup $S \subset G$, where $s^* = \tau(s)^{-1}$, 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 L\"uscher-Mack Theorem and the extensions of the L\"uscher-Mack Theorem for $*$-subsemigroups satisfying $S = S(G^\tau)_0$ by Merigon, Neeb, and \'Olafsson. Finally, we prove that non-degenerate strongly continuous representations of certain $*$-subsemigroups $S$ can even be extended to representations of a generalized version of an Olshanski semigroup.

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