Drinfeld Center and Representation Theory for Monoidal Categories
By Sergey Neshveyev, Makoto Yamashita. 2015
72 citations recorded by the source corpus.
Motivated by the relation between the Drinfeld double and central property (T) for quantum groups, given a rigid C*-tensor category $${\mathcal{C}}$$C and a unitary half-braiding on an ind-object, we construct a *-representation of the fusion algebra of $${\mathcal{C}}$$C. This allows us to present an alternative approach to recent results of Popa and Vaes, who defined C*-algebras of monoidal categories and introduced property (T) for them. As an example we analyze categories $${\mathcal{C}}$$C of Hilbert bimodules over a II1-factor. We show that in this case the Drinfeld center is monoidally equivalent to a category of Hilbert bimodules over another II1-factor obtained by the Longo–Rehren construction. As an application, we obtain an alternative proof of the result of Popa and Vaes stating that property (T) for the category defined by an extremal finite index subfactor $${N \subset M}$$N⊂M is equivalent to Popa’s property (T) for the corresponding SE-inclusion of II1-factors. In the last part of the paper we study Müger’s notion of weakly monoidally Morita equivalent categories and analyze the behavior of our constructions under the equivalence of the corresponding Drinfeld centers established by Schauenburg. In particular, we prove that property (T) is invariant under weak monoidal Morita equivalence.
Subfields: Geometry and Topology, Mathematical Physics
View Paper: https://www.semanticscholar.org/paper/af688ab311efa2fc563156b983505ae883bf8a55
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In Geometry and Topology
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Citations in the corpus
- Annular representation theory for rigid $C^{*}$-tensor categories (2015, background): “His result completely clarifies the relationship between this paper and the work of Neshveyev and Yamashita.”
- The monoidal center and the character algebra (2015, background): “ar tensor category, then its character table is obtained from the S-matrix (Example 6.14). Acknowledgments. The author thanks Victor Ostrik and Makoto Yamashita for letting him know about [Ost15] and [NY16], respectively, and giving him some 4 KENICHI SHIMIZU valuable comments. The author also thank”
- Cohomology and $L^2$-Betti numbers for subfactors and quasi-regular inclusions (2015, methodology): “ral rigid C∗-tensor categories C, we construct in Section 3.4 a canonical bijection between Hilbert space representations of the tube ∗-algebra A of C and unitary half braidings for C in the sense of [NY15a]. When C is a category of M-bimodules with corresponding SE-inclusion T⊂ S, we prove in Secti”
- Q-system completion for C⁎ 2-categories (2021, unknown): “Given a C ∗ / W ∗ 2-category C , its Q-system completion is the 2-category QSys ( C ) of Q-systems, bimodules, and intertwiners in C , which has been studied previously in [NY16, BKLR15, GY20], building on algebraic versions in [Yam04a, CR16, DR18, GJF19].”