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<p>We introduce the notion of a complex crown domain for a connected Lie group <i>G</i>, and we use analytic extensions of orbit maps of antiunitary representations to these domains to construct nets of real subspaces on <i>G</i> that are isotone, covariant and satisfy the Reeh–Schlieder and Bisognano–Wichmann conditions from Algebraic Quantum Field Theory. This provides a unifying perspective on various constructions of such nets. The representation theoretic properties of different crowns are discussed in some detail for the non-abelian 2-dimensional Lie group <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(\mathop {\textrm{Aff}}\nolimits ({\mathbb R})\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mtext>Aff</mtext> <mo stretchy="false">(</mo> <mi mathvariant="double-struck">R</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>. We also characterize the existence of nets with the above properties by a regularity condition in terms of an Euler element in the Lie algebra <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\({\mathfrak g}\)</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="fraktur">g</mi> </math></EquationSource> </InlineEquation> and show that all antiunitary representations of the split oscillator group have this property.</p>

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Crowned Lie groups and nets of real subspaces

  • 期刊论文

  • 发表日期：

    2026年7月6日

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  • Daniel Beltiţă,
  • Karl-Hermann Neeb

摘要

We introduce the notion of a complex crown domain for a connected Lie group G, and we use analytic extensions of orbit maps of antiunitary representations to these domains to construct nets of real subspaces on G that are isotone, covariant and satisfy the Reeh–Schlieder and Bisognano–Wichmann conditions from Algebraic Quantum Field Theory. This provides a unifying perspective on various constructions of such nets. The representation theoretic properties of different crowns are discussed in some detail for the non-abelian 2-dimensional Lie group \(\mathop {\textrm{Aff}}\nolimits ({\mathbb R})\) Aff ( R ) . We also characterize the existence of nets with the above properties by a regularity condition in terms of an Euler element in the Lie algebra \({\mathfrak g}\) g and show that all antiunitary representations of the split oscillator group have this property.