Crowned Lie groups and nets of real subspaces
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arXiv
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| Format: | Preprint |
| Published: |
2025
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| _version_ | 1866909653969928192 |
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| author | Beltita, Daniel Neeb, Karl-Hermann |
| author_facet | Beltita, Daniel Neeb, Karl-Hermann |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2506_16422 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Crowned Lie groups and nets of real subspaces Beltita, Daniel Neeb, Karl-Hermann Representation Theory 22E45, 30H10, 47B32, 47B91, 81R05, 81T05 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. |
| title | Crowned Lie groups and nets of real subspaces |
| topic | Representation Theory 22E45, 30H10, 47B32, 47B91, 81R05, 81T05 |
| url | https://arxiv.org/abs/2506.16422 |