  We construct doubled lattice Chern-Simons-Yang-Mills theories with discrete
gauge group $G$ in the Hamiltonian formulation. Here, these theories are
considered on a square spatial lattice and the fundamental degrees of freedom
are defined on pairs of links from the direct lattice and its dual,
respectively. This provides a natural lattice construction for
topologically-massive gauge theories, which are invariant under parity and
time-reversal symmetry. After defining the building blocks of the doubled
theories, paying special attention to the realization of gauge transformations
on quantum states, we examine the dynamics in the group space of a single
cross, which is spanned by a single link and its dual. The dynamics is governed
by the single-cross electric Hamiltonian and admits a simple quantum mechanical
analogy to the problem of a charged particle moving on a discrete space
affected by an abstract electromagnetic potential. Such a particle might
accumulate a phase shift equivalent to an Aharonov-Bohm phase, which is
manifested in the doubled theory in terms of a nontrivial ground-state
degeneracy on a single cross. We discuss several examples of these doubled
theories with different gauge groups including the cyclic group
$\mathbb{Z}(k)\subset U(1)$, the symmetric group $S_3\subset O(2)$, the binary
dihedral (or quaternion) group $\bar{D}_2\subset SU(2)$, and the finite group
$\Delta(27)\subset SU(3)$. In each case the spectrum of the single-cross
electric Hamiltonian is determined exactly. We examine the nature of the
low-lying excited states in the full Hilbert space, and emphasize the role of
the center symmetry for the confinement of charges. Whether the investigated
doubled models admit a non-Abelian topological state which allows for
fault-tolerant quantum computation will be addressed in a future publication.
