Extensions of a near-group category of type \((\mathbb{Z}_2;1)\). (English) Zbl 1513.18012
The theory of fusion categories arises in many areas of mathematics and physics such as semisimple Hopf algebras, quantum groups, vertex operator algebras and topological quantum field theory. Group extensions play a significant role in the theory of fusion categories. This paper is concerned with near-group categories of type \(\left( \mathbb{Z}_{2},1\right)\).
The principal objective in this paper is to establish the following two theorems.
I. Let \(\mathcal{D}\)be a near-group fusion category of type \(\left(\mathbb{Z}_{2},1\right) \)and \[\mathcal{C}={\bigoplus\nolimits_{g\in G}}\mathcal{C}_{g}\] be an extension of \(\mathcal{D}\). Assume that \(\mathcal{C}\)is braided. Then the structure of \(\mathcal{C}\) is exactly one of the following:
(1) If \(\mathcal{C}\) is of type \(\left( 1,2m;2,m\right) \), then \(\mathcal{C}\)is a \(\mathbb{Z}_{2}\)-equivariantization of a braided pointed fusion category \(\mathcal{D}\)of dimension \(3m\).
(2) If \(\mathcal{C}\) is of type \(\left( 1,2m;\sqrt{3},2m;2,m\right) \), then \(\mathcal{C}\)is a \(\mathbb{Z}_{2}\)-equivariantization of a fusion category \(\mathcal{D}\)which is of type \(\left( 1,3m;\sqrt{3},m\right) \) and is a \(\mathbb{Z}_{2}\)-extension of a pointed fusion category of dimension \(3m\). In particular, \(\mathcal{D}\)is not braided.
(3) If \(\mathcal{C}\) is of type \(\left( 1,2m;2,m;\sqrt{6},m\right) \), then \(\mathcal{C}\)is a \(\mathbb{Z}_{2}\)-equivariantization of a fusion category \(\mathcal{D}\)which is of type \(\left( 1,3m;\sqrt{6},m/2\right) \) and is a \(\mathbb{Z}_{2}\)-extension of a pointed fusion category of dimension \(3m\). In particular, \(\mathcal{D}\)is not braided.
II. Let \(\mathcal{D}\)be a near-group fusion category of type \(\left(\mathbb{Z}_{2},1\right) \)and \[\mathcal{C}={\bigoplus\nolimits_{g\in G}}\mathcal{C}_{g}\] be an extension of \(\mathcal{D}\). Then \(\mathcal{C}\) is group-theoretical in one of the following forms:
(1) \(\mathcal{C}\) is braided and integral.
(2) \(\mathcal{C}\) is equivalent as a tensor category to the category of finite-dimensional representations of semisimple Hopf algebra.
The principal objective in this paper is to establish the following two theorems.
I. Let \(\mathcal{D}\)be a near-group fusion category of type \(\left(\mathbb{Z}_{2},1\right) \)and \[\mathcal{C}={\bigoplus\nolimits_{g\in G}}\mathcal{C}_{g}\] be an extension of \(\mathcal{D}\). Assume that \(\mathcal{C}\)is braided. Then the structure of \(\mathcal{C}\) is exactly one of the following:
(1) If \(\mathcal{C}\) is of type \(\left( 1,2m;2,m\right) \), then \(\mathcal{C}\)is a \(\mathbb{Z}_{2}\)-equivariantization of a braided pointed fusion category \(\mathcal{D}\)of dimension \(3m\).
(2) If \(\mathcal{C}\) is of type \(\left( 1,2m;\sqrt{3},2m;2,m\right) \), then \(\mathcal{C}\)is a \(\mathbb{Z}_{2}\)-equivariantization of a fusion category \(\mathcal{D}\)which is of type \(\left( 1,3m;\sqrt{3},m\right) \) and is a \(\mathbb{Z}_{2}\)-extension of a pointed fusion category of dimension \(3m\). In particular, \(\mathcal{D}\)is not braided.
(3) If \(\mathcal{C}\) is of type \(\left( 1,2m;2,m;\sqrt{6},m\right) \), then \(\mathcal{C}\)is a \(\mathbb{Z}_{2}\)-equivariantization of a fusion category \(\mathcal{D}\)which is of type \(\left( 1,3m;\sqrt{6},m/2\right) \) and is a \(\mathbb{Z}_{2}\)-extension of a pointed fusion category of dimension \(3m\). In particular, \(\mathcal{D}\)is not braided.
II. Let \(\mathcal{D}\)be a near-group fusion category of type \(\left(\mathbb{Z}_{2},1\right) \)and \[\mathcal{C}={\bigoplus\nolimits_{g\in G}}\mathcal{C}_{g}\] be an extension of \(\mathcal{D}\). Then \(\mathcal{C}\) is group-theoretical in one of the following forms:
(1) \(\mathcal{C}\) is braided and integral.
(2) \(\mathcal{C}\) is equivalent as a tensor category to the category of finite-dimensional representations of semisimple Hopf algebra.
Reviewer: Hirokazu Nishimura (Tsukuba)
MSC:
| 18M20 | Fusion categories, modular tensor categories, modular functors |
| 16T05 | Hopf algebras and their applications |
Keywords:
fusion category; Hopf algebra; extension; near-group category; group-theoretical fusion categoryReferences:
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