Tube representations and twisting of graded categories

We study deformation of tube algebra under twisting of graded monoidal categories. When a tensor category \(\mathcal{C}\) is graded over a group \(\Gamma\), a torus-valued 3-cocycle on \(\Gamma\) can be used to deform the associator of \(\mathcal{C}\). Based on a natural Fell bundle structure of the tube algebra over the action groupoid of the adjoint action of \(\Gamma\), we show that the tube algebra of the twisted category is a 2-cocycle twisting of the original one.