  Kitaev's toric code is constructed using a finite gauge group from gauge
theory. Such gauge theories can be generalized with the gauge group generalized
to any finite-dimensional semisimple Hopf algebra. This also leads to
generalizations of the toric code. Here we consider the simple case where the
gauge group is unchanged but furnished with a non-trivial quasitriangular
structure (R-matrix), which modifies the construction of the gauge theory. This
leads to some interesting phenomena; for example, the space of functions on the
group becomes a non-commutative algebra. We also obtain simple Hamiltonian
models generalizing the toric code, which are of the same overall topological
type as the toric code, except that the various species of particles created by
string operators in the model are permuted in a way that depends on the
R-matrix. In the case of $\mathbb{Z}_{N}$ gauge theory, we find that the
introduction of a non-trivial R-matrix amounts to flux attachment.
