# How to make a premodular category a modular tensor category?

A premodular category (also called ribbon fusion category) is roughly speaking a tensor category where fusion and braiding of the objects are defined. With an extra nondegeneracy condition for the braiding, they are called modular tensor categories (MTCs) (cf. Chapter 4 of the book http://www.math.ucsb.edu/~zhenghwa/data/course/cbms.pdf).

I'd like to know, given a premodular category, is there a well-studied (minimal) way to complete it into an MTC? A simple example is to take the Toric code MTC (defined in page 39 of http://www.math.tamu.edu/~rowell/RSW.pdf), which consists of 4 objects $\{1,e,m,\epsilon \}$. The subcategory consists of $\{1,e\}$ is premodular because $e$ braids trivially with itself (or equivalently, the determinant of the left corner $2\times 2$ block of the S matrix is zero). We can add back the objects labeled $m$ and $\epsilon$, and stipulate that the topological spins are $\theta_{m} = 1$ and $\theta_{\epsilon}=-1$, respectively, and that the full braiding between $e$ and $m$, $\epsilon$ and $e$, $\epsilon$ and $m$ are all -1. In this way, we complete the premodular category into an MTC. Note that we also need to defined the fusion rules and $F$ symbols of the new category in a consistent way (satisfying the pentagon equations), and that they should play well with the braidings statistics (satisfying the hexagon equations).

The case I'm particularly interested in is when the premodular category is the representation category of finite groups (or quantum groups). As a physicist, I like to think about the toric code example above as adding back the $\mathbb{Z}_2$ flux ($m$) to the $\mathbb{Z}_2$ charge ($e$) (so that the two braid nontrivially) and the consistency of the new category requires us also to add the charge-flux composite $\epsilon$. I'm not sure if I can generalize this procedure to the representation category of groups (or quantum groups), in which case I can presumably think of the objects in the premodular category as charges in a gauge theory.

A premodular category is always spherical, so you can take the Drinfel'd center:

http://arxiv.org/pdf/math/0111205v1.pdf

EDIT: I probably should have pointed out, as Marcel does in the comments, "that the original category naturally embeds as a braided category into its center."

• Note the Drinfel'd center is a generalization of the Drinfel'd quantum double of a finite group. The toric code UMTC is precisely the quantum double of Z/2. – Eric S. Jan 22 '16 at 6:21
• And note that the original category naturally embeds as a braided category into its center. – Marcel Bischoff Jan 22 '16 at 14:57
• Hi Eric, Thanks! This is exactly what I need. I didn't realize that the original category can be naturally embedded into its center as Marcel mentioned in the previous thread. – Zitao Wang Jan 28 '16 at 4:21

Whether there is a "minimal" way to do this is still open, as far as I understand. One concrete formulation is in Mueger's "On the structure of modular categories" (Conjecture 5.2), see http://arxiv.org/abs/math/0201017.