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We consider lattice Hamiltonian realizations of ($d$+1)-dimensionalDijkgraaf-Witten theory. In (2+1)d, it is well-known that the Hamiltonianyields point-like excitations classified by irreducible representations of thetwisted quantum double. This can be confirmed using a tube algebra approach. Inthis paper, we propose a generalization of this strategy that is valid in anydimensions. We then apply the tube algebra approach to derive the algebraicstructure of loop-like excitations in (3+1)d, namely the twisted quantumtriple. The irreducible representations of the twisted quantum triple algebracorrespond to the simple loop-like excitations of the model. Similarly to its(2+1)d counterpart, the twisted quantum triple comes equipped with a compatiblecomultiplication map and an $R$-matrix that encode the fusion and the braidingstatistics of the loop-like excitations, respectively. Moreover, we explainusing 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 $n$-lowerdimensions. This can in turn be used to recast higher-dimensional tube algebrasin terms of lower dimensional analogues.

Tube algebras, excitations statistics and compactification in gauge models of topological phases