Weakly group-theoretical and solvable fusion categories

Etingof, Pavel; Nikshych, Dmitri; Ostrik, Viktor

doi:10.1016/j.aim.2010.06.009

Cited by 183 publications

(345 citation statements)

References 43 publications

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“…The notion of solvability for semisimple Hopf algebras was introduced by Etingof et al [5]. A semisimple Hopf algebra is called solvable if the category of its finite dimensional representations is a solvable fusion category.…”

Section: In Particular D(h) Has Irreducible Modules Of Dimension 2 Amentioning

confidence: 99%

“…However, the interrelation between solvability and semisolvability for semisimple Hopf algebras is not clear enough. For example, Etingof et al proved [5,Theorem 1.6] that semisimple Hopf algebras of dimension p a q b are…”

Section: In Particular D(h) Has Irreducible Modules Of Dimension 2 Amentioning

confidence: 99%

“…See [16,5,20] for details. Our present work is devoted to providing a new class of semisimple Hopf algebras which are semisolvable.…”

Section: Introductionmentioning

confidence: 99%

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Semisimple Hopf Algebras of Dimension 2q³

Dong

Dai

2015

Mathematica Slovaca

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Section: In Particular D(h) Has Irreducible Modules Of Dimension 2 Amentioning

confidence: 99%

Section: In Particular D(h) Has Irreducible Modules Of Dimension 2 Amentioning

confidence: 99%

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Semisimple Hopf Algebras of Dimension 2q³

Dong

Dai

2015

Mathematica Slovaca

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“…Let M be a left semisimple A-module category, and let N be a right semisimple A-module category. Consider the tensor product N ⊠ A M (see [ENO2]). Namely, if A 1 , A 2 are algebras in A such that M = mod − A 1 and N = A 2 − mod, then N ⊠ A M is the category of (A 2 , A 1 )-bimodules in A, which can also be described as the category of left A 2 -modules in M, or the category of right A 1 -modules in N (see [EGNO,Section 7.8]).…”

Section: Preliminariesmentioning

confidence: 99%

Exact factorizations and extensions of fusion categories

Gelaki

2017

Journal of Algebra

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Abstract. We introduce and study the new notion of an exact factorization B = A • C of a fusion category B into a product of two fusion subcategories A, C ⊆ B of B. This is a categorical generalization of the well known notion of an exact factorization of a finite group into a product of two subgroups. We then relate exact factorizations of fusion categories to exact sequences of fusion categories with respect to an indecomposable module category, which was introduced and studied in [EG]. We also apply our results to study extensions of a group-theoretical fusion category by another one, provide some examples, and propose a few natural questions.

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“…Recall ( [DGNO], [ENO2]) that a fusion category is called weakly group-theoretical if it is obtained by a chain of extensions and equivariantizations from the trivial category. Proof.…”

Section: Good and Bad Primesmentioning

confidence: 99%

Reductions of Tensor Categories Modulo Primes

Etingof

Gelaki

2011

Communications in Algebra

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Abstract. We study good (i.e., semisimple) reductions of semisimple rigid tensor categories modulo primes. A prime p is called good for a semisimple rigid tensor category C if such a reduction exists (otherwise, it is called bad). It is clear that a good prime must be relatively prime to the Müger squared norm |V | 2 of any simple object V of C. We show, using the Ito-Michler theorem in finite group theory, that for group-theoretical fusion categories, the converse is true. While the converse is false for general fusion categories, we obtain results about good and bad primes for many known fusion categories (e.g., for Verlinde categories). We also state some questions and conjectures regarding good and bad primes.

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Weakly group-theoretical and solvable fusion categories

Cited by 183 publications

References 43 publications

Semisimple Hopf Algebras of Dimension 2q³

Semisimple Hopf Algebras of Dimension 2q³

Exact factorizations and extensions of fusion categories

Reductions of Tensor Categories Modulo Primes

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Weakly group-theoretical and solvable fusion categories

Cited by 183 publications

References 43 publications

Semisimple Hopf Algebras of Dimension 2q3

Semisimple Hopf Algebras of Dimension 2q3

Exact factorizations and extensions of fusion categories

Reductions of Tensor Categories Modulo Primes

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Semisimple Hopf Algebras of Dimension 2q³

Semisimple Hopf Algebras of Dimension 2q³