Non-group-theoretical semisimple Hopf algebras from group actions on fusion categories

By D. Nikshych. 2007

82 citations recorded by the source corpus.

Abstract.Given an action of a finite group G on a fusion category $${\mathcal{C}}$$ we give a criterion for the category of G-equivariant objects in $${\mathcal{C}}$$ to be group-theoretical, i.e., to be categorically Morita equivalent to a category of group-graded vector spaces. We use this criterion to answer affirmatively the question about existence of non-group-theoretical semisimple Hopf algebras asked by P. Etingof, V. Ostrik, and the author in [7]. Namely, we show that certain $${\mathbb{Z}}$$/2$${\mathbb{Z}}$$-equivariantizations of fusion categories constructed by D. Tambara and S. Yamagami [26] are equivalent to representation categories of non-group-theoretical semisimple Hopf algebras. We describe these Hopf algebras as extensions and show that they are upper and lower semisolvable.

Subfields: Algebra and Number Theory, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/cc0f0968c7b8754801df77bd04e082868dc286ad

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Citations in the corpus

Centers of graded fusion categories (2009, methodology): “e a series of non group-theoretical fusion categories admitting ﬁber functors (i.e., representation categories of non group-theoretical semisimple Hopf algebras), generalizing examples constructed in [Ni]. Let Abe a ﬁnite Abelian group with a non-degenerate bilinear form χ. Let Aut(A,χ) denote the g”
On finite-dimensional Hopf algebras (2014, background): “123 Notice that there are semisimple extensions that are not abelian [40, 69, 74].”
Exact sequences of tensor categories (2010, background): “In that case, it is shown in [32] that C is dual to a crossed product fusion category C ⋊G with respect to the indecomposable module category C.”
A Finiteness Property for Braided Fusion Categories (2009, background): “…integral braided fusion categories in the literature: Drinfeld centers of Tambara-Yamagami categories DT Y(A, χ, τ) (see [19, 17] and Section 5 below), and quantum group type modular categories C(soN , q, ℓ) where ℓ = N or 2N if N is even or odd respectively (see e.g. [15] and Section 3 below).”