  In the present paper we study tensor C*-categories with non-simple unit
realised as C*-dynamical systems (F,G,\beta) with a compact (non-Abelian) group
G and fixed point algebra A := F^G. We consider C*-dynamical systems with
minimal relative commutant of A in F, i.e. A' \cap F = Z, where Z is the center
of A which we assume to be nontrivial. We give first several constructions of
minimal C*-dynamical systems in terms of a single Cuntz-Pimsner algebra
associated to a suitable Z-bimodule. These examples are labelled by the action
of a discrete Abelian group (which we call the chain group) on Z and by the
choice of a suitable class of finite dimensional representations of G. Second,
we present a construction of a minimal C*-dynamical system with nontrivial Z
that also encodes the representation category of G. In this case the C*-algebra
F is generated by a family of Cuntz-Pimsner algebras, where the product of the
elements in different algebras is twisted by the chain group action. We apply
these constructions to the group G = SU(N).
