Projective objects and the modified trace in factorisable finite tensor categories

Understand

For C a factorisable and pivotal finite tensor category over an algebraically closed field of characteristic zero we show: 1) C always contains a simple projective object; 2) if C is in addition ribbon, the internal characters of projective modules span a submodule for the projective SL(2,Z)-action; 3) the action of the Grothendieck ring of C on the span of internal characters of projective objects can be diagonalised; 4) the linearised Grothendieck ring of C is semisimple iff C is semisimple.

• Results 1-3 remain true in positive characteristic under an extra assumption.
• Result 1 implies that the tensor ideal of projective objects in C carries a unique-up-to-scalars modified trace function.
• We express the modified trace of open Hopf links coloured by projectives in terms of S-matrix elements.