AG Lie-Gruppen: K.-H. Neeb, FAU: Nets of Local Algebras on Causal Flag Manifolds

Nets of Local Algebras
on Causal Flag Manifolds – Vortragender: Karl-Hermann Neeb, Friedrich-Alexander-Universität Erlangen-Nürnberg

Abstract: We are interested in obtaining local nets in
the sense of Haag–Kastler from unitary representations of a connected Lie
group G. A natural set of axioms leads to a causal structure on M. We focus on
the case where M = G/P is a flag manifold of a simple Lie group G, or a
covering space thereof. Then G must be a hermitian Lie group and M a conformal
compactification of a euclidean Jordan algebra V. Its simply connected covering
is a simple space-time manifold in the sense of Mack–de Riese.

We show that the unitary representations permitting
non-trivial nets are the positive energy representations (direct integrals of
lowest weight representations). These nets have several interesting features. One
is that locality properties of the net can be specified in terms of open
G-orbits in the space of pairs, which is most interesting for covering spaces
because the number of these orbits corresponds to the number of sheets in the
covering.