  This article presents a precise description of the interplay between the
symmetries of a quantum or classical theory with spacetime interpretation, and
some of its physical properties relating to causality, horizons and positive
energy. Our major result is that the existence of static metrics on spacetimes
and that of positive energy representations of symmetry groups, are equivalent
to the existence of particular Adjoint-invariant convex cones in the symmetry
algebras. This can be used to study backgrounds of supergravity and string
theories through their symmetry groups. Our formalism is based on Segal's
approach to infinitesimal causal structures on manifolds. The Adjoint action in
the symmetry group is shown to correspond to changes of inertial frames in the
spacetime, whereas Adjoint-invariance encodes invariance under changes of
observers. This allows us to give a group theoretical description of the
horizon structure of spacetimes, and also to lift causal structures to the
Hilbert spaces of quantum theories. Among other results, by setting up the
Dirac procedure for the complexified universal algebra, we classify the
physically inequivalent observables of quantum theories. We illustrate this by
finding the different Hamiltonians for stationary observers in AdS_2.
