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.



Drinfeld center of a braided category
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.


