Question

Suppose that C is a fusion category (over the complex numbers) and that Z(C) is its Drinfel'd center. By definition an object in Z(C) consists of an object V in C together with a collection of half-braidings $V \otimes W \rightarrow W \otimes V$ for every object W in C satisfying some naturality conditions. Hence there is a restriction functor R:Z(C)->C given by forgetting the half-braiding. Adjoint to this is an induction functor I:C->Z(C).

A Theorem of Etingof-Nikshych-Ostrik says that $R(I(V)) = \bigoplus_X X \otimes V \otimes X^{*}$. In particular, we see that I(V) is $\bigoplus_X X \otimes V \otimes X^{*}$ (where X ranges over simple objects up to isomorphism) together with some particular choice of half-braidings. I'm pretty sure I know what those half-braidings are. In particular there's a nice picture (the X,Y summand of the half-braiding with W is a sum over diagrams with a trivalent vertex connecting W to X* and Y* and another trivalent vertex connecting X and Y to W, where the two vertices range over dual bases).

What I really would like is a reference that explains this so that I don't have to write it up myself. The only description I know is in ENO's "On Fusion Categories" where it's written in terms of weak Hopf algebras.

The motivation is removing any mention of weak Hopf algebras from the construction in Section 5 (about cyclotomicity of certain Drinfel'd centers) in Scott and my Noncyclotomic Fusion Categories. It turns out that the diagram description above can be slightly modified (in a way suggested to me by Ben Webster) in order to give a description of I(V) where V is an object in a non-split fusion category over an arbitrary field.

Answer 1

In the meanwhile a reference has appeared: Theorem 2.3 in http://arxiv.org/abs/1004.1533

Answer 2

I suggest trying this paper of Mueger: http://arxiv.org/abs/math/0111205. In section 3.3 he defines a map $Hom_{C}(X,Y) \rightarrow Hom_{Z(C)}((X,e_X),(Y,e_Y))$ (actually a trace preserving conditional expectation) which, as far as I understand, is what you are looking for. It implies $R(I(V)) = \oplus_X X \otimes V \otimes X^* $ almost tautologically, I would say.

If I remember well, Mueger assumed $dim C$ to be non-zero, but this turned out to be always true (proven in the paper of Etingof-Nykshych-Ostrik).

Edit I read your question again, and now I see which half-braiding you mean. Proof and diagrams are almost the same as the ones used to show that $\oplus_X X \boxtimes X^{op}$ is a Frobenius algebra in $C \boxtimes C^{op}$. I doubt it has been written in detail anywhere.

Concerning the theorem, I don't know if the following is helpful. Denote $Q = \oplus_X X \boxtimes X^{op}$. $Z(C)$ is equivalent to the tensor category of $Q-Q$-bimodules. For $V \in C$, $I(V)$ corresponds to $Q \otimes (V \boxtimes 1) \otimes Q$. Notice that in general $(V^* \boxtimes 1) \otimes Q \sim (1 \boxtimes V^{op}) \otimes Q$ and there is a canonical choice for this isomorphism.

For $(Y,e_Y) \in Z(C)$, $(Y \boxtimes 1) \otimes Q$ has the structure of a $Q-Q$-bimodule: let $Q$ act on the right the obvious way, and use the half braiding $e_y$ to let $Q$ act from the left as well. All $Q-Q$ bimodules are of this form (up to isomorphism). Thus the restriction functor sends $(Y \boxtimes 1) \otimes Q$ to $Y$.

As you mentioned, $\oplus_X X \otimes V \otimes X^*$ has a natural choice of half braiding: choose a basis for each $Hom_C(X \otimes Y, Z)$, where $X,Y,Z$ span a complete set of irreducibles, and use them (together with the rigidity structure) to build an isomorphism $ (\oplus_X X \otimes V \otimes X^*) \otimes Y \sim Y \otimes (\oplus_Z Z \otimes V \otimes Z^*)$ satisfying the necessary conditions.

Or define it through this sequence of isomorphims (so maybe you can avoid diagrams, in the end): $$ (Y \boxtimes 1) \otimes ((\oplus_X X \otimes V \otimes X^*) \boxtimes 1) \otimes Q \sim (Y \boxtimes 1) \otimes (\oplus_X (X \otimes V ) \boxtimes X^{op}) \otimes Q$$

$$\sim (Y \boxtimes 1) \otimes Q \otimes ( V \boxtimes 1) \otimes Q \sim (1 \boxtimes Y^{op *}) \otimes ((\oplus_X X \otimes V \otimes X^*) \boxtimes 1) \otimes Q$$

$$\sim ((\oplus_X X \otimes V \otimes X^*) \boxtimes 1) \otimes ( 1 \boxtimes Y^{op *}) \otimes Q \sim ((\oplus_X X \otimes V \otimes X^*) \boxtimes 1) \otimes (Y \boxtimes 1) \otimes Q.$$

Thus $((\oplus_X X \otimes V \otimes X^*) \boxtimes 1) \otimes Q \sim Q \otimes (V \boxtimes 1) \otimes Q$ not only as objects in $ C \boxtimes C^{op}$, but also as $Q-Q$-bimodules (and $(R(I(V)) \sim \oplus_X X \otimes V \otimes X^*$).