Drinfeld center of a braided category Suppose I have a braided monoidal category $\mathcal{C}$. I specifically am interested in the case where $\mathcal{C}$ is the category of finite-dimensional modules of a quantum group, say $\mathcal{U}_q(\mathfrak{sl}_2)$ (or a variant of it.)
The braiding $c_{-,-}$ embeds $\mathcal{C}$ into its Drinfeld center $\mathcal{Z}(\mathcal{C})$ via
$$ V \mapsto (V, c_{V,-} ). $$
Does this give the entire Drinfeld center? If not, is it easy to see what parts of $\mathcal{Z}(\mathcal{C})$ it misses, at least in this case?
Is there a reference that discusses this? I think this should be related to a theorem of the form $D(D(H)) \cong D(H)$ (where $D(H)$ is the Drinfeld double of a Hopf algebra) but I don't recall a reference for that result either.
 A: No, the functor $\mathcal C \to \mathcal Z(\mathcal C)$ is not essentially surjective in general. 
For example, in the case you have in mind, $\mathcal C = Rep_q(G)$ (say $G$ a semisimple algebraic group), the Drinfeld center can be identified with the category 
$HC_q := \mathcal O^{RE}_q(G)-mod_{Rep_q(G)}$
of modules for the so-called reflection equation algebra $\mathcal O_q^{RE}(G)$ internal to $Rep_q(G)$.
The image of $Rep_q(G)$ in thus identified with those modules on which the REA acts trivially (i.e. via the augmentation $\varepsilon: \mathcal O_q^{RE}(G) \to \mathbb C$).
Note that this holds even when $q=1$ (and so $Rep(G)$ is symmetric monoidal). Then $HC_{q=1}$ is the same thing as $Coh(G/G)$, the category of $G$-equivariant coherent sheaves on $G$. The image of $Rep(G)$ consists of coherent sheaves supported on the identity element of $G$. This example also makes sense for a finite group. 
Note also that the Drinfeld center may be non-symmetrically braided even when $\mathcal C$ is symmetric.
