Crowned Lie groups and nets of real subspaces

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.