If C is a fusion category and $\dim(C) \neq 0$ (the latter is automatic in characteristic zero, but not in nonzero characteristic), then the Drinfel'd center Z(C) is fusion. More generally, if C is a fusion category and M is a semisimple module category over C, then the dual of C over M is fusion if dim(C) is nonzero. But if $\dim(C)=0$ these results need not hold. For example Vec(G) in characteristic p when $p \mid \#G$ acts on Vec and its dual is Rep(G) which is not semisimple. Similarly, Z(Vec(G)) is not semisimple when $p \mid \#G$.
I want to know if C and Z(C) both being semisimple implies that $\dim(C) \neq 0$.
(Feel free to assume everywhere that the base field is algebraically closed if you'd like to.)