# The Modular Closure of Braided Tensor Categories

• Michael Müger
Conference paper
Part of the Lecture Notes in Physics book series (LNP, volume 543)

## Abstract

With every monoid M one can associate its center Z(M)={xM|xy=yxyM}, which obviously is a commutative monoid. If M has inverses, i.e. is a group, then Z(M) is a normal subgroup and one can consider the quotient group M|Z(M). In nice cases, e.g. if M is a direct product of simple groups, M|Z(M) turns out to have trivial center. Monoids being 0-categories, the work to be reported here can be considered as the analogous construction for 1-categories. We refer to Müger (1998) for a full account. Given a strict tensor category ∋ with braiding ε, we define its center to be the full subcategory defined by
$$ObjZ\left( \mathcal{C} \right) = \left\{ {\rho \in Obj\mathcal{C}\left| {\varepsilon \left( {\rho ,\sigma } \right)o\varepsilon \left( {\sigma ,\rho } \right) = id_{\sigma \otimes \rho } \forall \sigma \in Obj\mathcal{C}} \right.} \right\},$$
, which clearly is a symmetric tensor category.

## References

1. M. Müger (1998): The modular closure of braided tensor categories. (math. CT/9812040) To appear in Adv. Math.Google Scholar
2. M. Müger (1999): On Galois extensions of braided tensor categories, mapping class group representations and simple current extensions. In preparationGoogle Scholar
3. S. Doplicher and J. E. Roberts (1989): A new duality theory for compact groups. Invent. Math. 98, 157–218