Is every separable algebra in a modular tensor category Morita equivalent to a commutative one? Separable algebras in modular tensor categories are interesting algebraic structures, which have received significant attention because of their connection to conformal field theories. My understanding is that it is only the Morita class of the algebra which is important for determining the conformal field theory. Also, in the more general case of a fusion category, it is known by a result of Ostrik that up to Morita equivalence, separable algebras correspond to module categories for the fusion category.
Over ordinary vector spaces, every separable algebra is Morita equivalent to a commutative one, which of course makes the Morita equivalence classes easier to study. So this motivates my question: is every separable algebra in a modular tensor category Morita equivalent to a commutative one? If not, what about restricting to symmetric separable algebras?
I am working always over $\mathbb{C}$ here, and assuming that the categories involved are semisimple and with a finite number of isomorphism classes of simple objects.
 A: The answer to your first question is no.  The category of modules over a commutative algebra has a tensor category structure such that the forgetful functor is a tensor functor, but the category of modules over noncommutative algebras need not.  Since the category of modules is a Morita invariant this gives a negative answer.  The usual name for this kind of situation is "Type II modular invariant" and the D_odd module categories over su(2) at roots of unity are the simplest examples.  See Kirillov-Ostrik for more details on everything in this paragraph.  In these examples you have a copy of super vector spaces inside su(2) at a root of unity and the algebra is the anti-symmetric one.  
("Morally" super vector spaces with the group ring of Z/2 with each group element supported in its grade is the simplest answer to your question, but technically it's not modular.  But you can find it inside modular categories.)
It is worth noting that module categories over a modular tensor category do have a classification in terms of commutative algebras.  Namely, you pick two commutative algebras A and B plus a braided equivalence between $\mathrm{Rep}^0(A)$ and $\mathrm{Rep}^0(B)$.  This is proved in Davydov-Nikshych-Ostrik Cor 3.8, though the idea goes back at least to Ocneanu.
