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Central exact sequences of tensor categories, equivariantization and applications. (English) Zbl 1311.18009
This article is built on the previous work of the authors [Int. Math. Res. Not. 2011, No. 24, 5644–5705 (2011; Zbl 1250.18005)]. The main idea of this sequence of works is to obtain analogues for tensor categories of the relation between exact sequences of groups, or more generally Hopf algebras, and extensions of these objects. The main result of this paper characterizes the exact sequences of (finite) tensor categories that correspond to equivariantizations, which are an important tool in constructions and classifications of fusion categories.
An exact sequence $$\mathcal{C}' \longrightarrow \mathcal{C} \overset{F}{\longrightarrow} \mathcal{C}''$$ of tensor categories is an equivariantization exact sequence, if it is equivalent to one of the form $$\mathrm{rep}(G) \longrightarrow (\mathcal{C}'')^{G} \longrightarrow \mathcal{C}''$$, where $$G$$ is a finite group scheme that acts on $$\mathcal{C}''$$ and $$(\mathcal{C}'')^{G}$$ is the corresponding equivariantization.
By considering the adjoint of the functor $$F$$, an exact sequence induces an algebra in $$\mathcal{C}$$ that lifts to an algebra in the center $$\mathcal{Z}(\mathcal{C})$$ and the sequence is called central if the two categories of modules are equivalent under the forgetful functor $$U: \mathcal{Z}(\mathcal{C}) \longrightarrow \mathcal{C}$$. Central exact sequences are shown to be equivalent to an associated normal Hopf monad being exact and cocommutative and to equivariantization exact sequences. The authors give two proofs of this statement and thereby connect to yet another characterization of equivariantization in [P. Etingof et al., Adv. Math. 226, No. 1, 176–205 (2011; Zbl 1210.18009)].
As applications, central exact sequences for Tambara-Yamagami categories are considered and the authors give sufficient criteria for recognizing tensor functors that fit into an equivariantization exact sequence.

##### MSC:
 18D10 Monoidal, symmetric monoidal and braided categories (MSC2010) 16T05 Hopf algebras and their applications
##### Citations:
Zbl 1250.18005; Zbl 1210.18009
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##### References:
 [1] A. Bruguières, Catégories prémodulaires, modularisations et invariants des variétés de dimension 3, Math. Ann., 316 (2000), 215-236. · Zbl 0943.18004 [2] A. Bruguières, S. Lack and A. Virelizier, Hopf monads on monoidal categories, Adv. Math., 227 (2011), 745-800. · Zbl 1233.18002 [3] A. Bruguières and S. Natale, Exact sequences of tensor categories, Int. Math. Res. Not. IMRN, 2011 (2011), 5644-5705. · Zbl 1250.18005 [4] A. Bruguières and A. Virelizier, Quantum double of Hopf monads and categorical centers, Trans. Amer. Math. Soc., 364 (2012), 1225-1279. · Zbl 1288.18004 [5] V. Drinfeld, S. Gelaki, D. Nikshych and V. Ostrik, On braided fusion categories I, Selecta Math. (N.S.), 16 (2010), 1-119. · Zbl 1201.18005 [6] P. Etingof, D. Nikshych and V. Ostrik, On fusion categories, Ann. of Math. (2), 162 (2005), 581-642. · Zbl 1125.16025 [7] P. Etingof, D. Nikshych and V. Ostrik, Weakly group-theoretical and solvable fusion categories, Adv. Math., 226 (2011), 176-205. · Zbl 1210.18009 [8] S. Gelaki and D. Nikshych, Nilpotent fusion categories, Adv. Math., 217 (2008), 1053-1071. · Zbl 1168.18004 [9] T. Kobayashi and A. Masuoka, A result extended from groups to Hopf algebras, Tsukuba J. Math., 21 (1997), 55-58. · Zbl 0894.16019 [10] M. Müger, Galois theory for braided tensor categories and the modular closure, Adv. Math., 150 (2000), 151-201. · Zbl 0945.18006 [11] M. Müger, On the structure of modular categories, Proc. London Math. Soc. (3), 87 (2003), 291-308. · Zbl 1037.18005 [12] S. Natale, On group theoretical Hopf algebras and exact factorizations of finite groups, J. Algebra, 270 (2003), 199-211. · Zbl 1040.16027 [13] S. Natale, Semisolvability of Semisimple Hopf Algebras of Low Dimension, Mem. Amer. Math. Soc., 186 , Amer. Math. Soc., Providence RI, 2007. · Zbl 1185.16033 [14] B. Pareigis, On braiding and dyslexia, J. Algebra, 171 (1995), 413-425. · Zbl 0816.18003 [15] P. Schauenburg, The monoidal center construction and bimodules, J. Pure Appl. Algebra, 158 (2001), 325-346. · Zbl 0984.18006 [16] H.-J. Schneider, Some remarks on exact sequences of quantum groups, Comm. Algebra, 21 (1993), 3337-3357. · Zbl 0801.16040 [17] D. Tambara and S. Yamagami, Tensor categories with fusion rules of self-duality for finite abelian groups, J. Algebra, 209 (1998), 692-707. · Zbl 0923.46052
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