Crowned Lie groups and nets of real subspaces

2025/06/19 by Beltita, Daniel, Neeb, Karl-Hermann
#22E45 #30H10 #47B32 #47B91 #81R05 #81T05 #FOS: Mathematics #Representation Theory (math.RT)

paper · doi:10.48550/arxiv.2506.16422

Abstract

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 \rm Aff(\mathbb 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 and show that all antiunitary representations of the split oscillator group have this property.

Citations

• Nets of real subspaces on homogeneous spaces and Algebraic Quantum Field Theory
• Orthogonal pairs of Euler elements I. Classification, fundamental groups and twisted duality
• Polynomial growth of holomorphic extensions of orbit maps of K-finite vectors at the boundary of the crown
• Realization of unitary representations of the Lorentz group on de Sitter space
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• Holomorphic extension of one-parameter operator groups
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• Nets of standard subspaces on Lie groups
• Antiunitary representations and modular theory
• On Differentiable Vectors for Representations of Infinite Dimensional\n Lie Groups
• Lie groups and Lie algebras

Related

• Nets of real subspaces on homogeneous spaces and Algebraic Quantum Field Theory
• Nets of standard subspaces on Lie groups
• From local nets to Euler elements
• Méthode des Orbites et Représentations Quantiques
• Nets of standard subspaces induced by antiunitary representations of admissible Lie groups I