List of papers with comments

1.  Mizuki Oikawa. Frobenius algebras associated with the α-induction for equivariantly braided tensor categories. Ann. Henri Poincaré, 25(10): 4423–4495, 2024. Springer Link arXiv:2303.11845

Abstract: Let G be a group. We give a categorical definition of the G-equivariant α-induction associated with a given G-equivariant Frobenius algebra in a G-braided multitensor category, which generalizes the α-induction for G-twisted representations of conformal nets. For a given G-equivariant Frobenius algebra in a spherical G-braided fusion category, we construct a G-equivariant Frobenius algebra, which we call a G-equivariant α-induction Frobenius algebra, in a suitably defined category called neutral double. This construction generalizes Rehren's construction of α-induction Q-systems. Finally, we define the notion of the G-equivariant full center of a G-equivariant Frobenius algebra in a spherical G-braided fusion category and show that it indeed coincides with the corresponding G-equivariant α-induction Frobenius algebra, which generalizes a theorem of Bischoff, Kawahigashi and Longo.

Comment: The first observation was that the G-equivariant version of the conjugation of morphisms (Definition 4.9) works if we replace Deligne tensor products by crossed products (Proposition 4.1), which was in November 2021. Then, many things went well and I obtained the proof of the main theorems for conformal nets in February 2022, but I tried to reformulate the definition of equivariant α-induction and generalize the theorems in the setting where categories and group actions are non-strict in order to apply them also to vertex  operator algebras (VOAs), which took me over one year.  Especially, it took me long to understand coherence theorems.

2.  Center construction for group-crossed tensor categories. arXiv preprint arXiv:2404.09972

Abstract: We define the notion of the (G, Γ)-crossed center of a (G, Γ)-crossed tensor category in the sense of Natale. We show that the (G, Γ)-crossed center is a (G⋈Γ, G×Γ)-braided tensor category. This construction generalizes the graded center construction for graded tensor categories and the equivariant center construction for tensor categories with group actions.

Comment: It was useful that I found Natale's work in MathSciNet by chance. Because the group action of a (G, Γ)-crossed tensor category does not preserve the tensor product in the ordinary sense, if I had not known Natale's work, possibly I would have concluded that a (G, Γ)-crossed center would not admit a natural group action. 

The title of my Ph. D. thesis is

Equivariant α-induction Frobenius algebras and related constructions of tensor categories,

which is based on the article [1].