  We consider lattice Hamiltonian realizations of ($d$+1)-dimensional
Dijkgraaf-Witten theory. In (2+1)d, it is well-known that the Hamiltonian
yields point-like excitations classified by irreducible representations of the
twisted quantum double. This can be confirmed using a tube algebra approach. In
this paper, we propose a generalization of this strategy that is valid in any
dimensions. We then apply the tube algebra approach to derive the algebraic
structure of loop-like excitations in (3+1)d, namely the twisted quantum
triple. The irreducible representations of the twisted quantum triple algebra
correspond to the simple loop-like excitations of the model. Similarly to its
(2+1)d counterpart, the twisted quantum triple comes equipped with a compatible
comultiplication map and an $R$-matrix that encode the fusion and the braiding
statistics of the loop-like excitations, respectively. Moreover, we explain
using the language of loop-groupoids how a model defined on a manifold that is
$n$-times compactified can be expressed in terms of another model in $n$-lower
dimensions. This can in turn be used to recast higher-dimensional tube algebras
in terms of lower dimensional analogues.
