TUBE REPRESENTATIONS AND TWISTING OF GRADED CATEGORIES

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