Journal of Lie Theory 32 (2022), No. 1, 029--074
Copyright Heldermann Verlag 2022
Nets of Standard Subspaces Induced by Antiunitary Representations of Admissible Lie Groups I
Daniel Oeh
Department Mathematik, Friedrich-Alexander-Universität Erlangen-Nürnberg, Erlangen, Germany
oehd@math.fau.de
[Abstract-pdf]
\newcommand{\cH}{\mathcal{H}} \newcommand{\g}{{\mathfrak g}} \renewcommand{\L}{\mathop{\bf L{}}\nolimits} \newcommand{\1}{\mathbf{1}} \newcommand{\ad}{\mathop{{\rm ad}}\nolimits} Let \((\pi, \cH)\) be a strongly continuous unitary representation of a 1-connected Lie group \(G\) such that the Lie algebra \(\g\) of \(G\) is generated by the positive cone \(C_\pi := \{x \in \g : -i\partial \pi(x) \geq 0\}\) and an element \(h\) for which the adjoint representation of \(h\) induces a 3-grading of \(\g\). Moreover, suppose that \((\pi, \cH)\) extends to an antiunitary representation of the extended Lie group \(G_\tau := G \rtimes \{\1, \tau_G\}\), where \(\tau_G\) is an involutive automorphism of \(G\) with \(\L(\tau_G) = e^{i\pi\ad h}\). In a recent work by Neeb and {\'Olafsson}, a method for constructing nets of standard subspaces of \(\cH\) 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 \(\g_C = C_\pi - C_\pi \subset \g\) is one-dimensional. The case where the center of \(\g_C\) has more than one dimension will be discussed in a separate paper.
Keywords: Standard subspace, Lie group, covariant net, quantum field theory.
MSC: 22E45, 81R05, 81T05.
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