Fiber Functors of Equivariantizations of Finite Tensor Categories

Abstract: Let $G$ be a finite group acting on a finite tensor category $\mathcal{C}$. We classify fiber functors on the equivariantization $\mathcal{C}<sup>G$ in terms of equivariant exact module categories over $\mathcal{C}$, indexed by subgroups of $G$. The data are a subgroup $H\subseteq G$ and an $H$-equivariant $\mathcal{C}$-module category $\mathcal{M}$ whose underlying $\mathcal{C}$-module category is indecomposable, exact, and semisimple; they give a fiber functor precisely when $H$ acts transitively on the simple objects of $\mathcal{M}$ and the stabilizer cocycle of one, hence every, simple object is non-degenerate. Through Tannaka-Krein reconstruction this describes realizations of $\mathcal{C}<sup>G$ as the representation category of a finite-dimensional Hopf algebra, with no semisimplicity hypothesis on $\mathcal{C}$. As applications, for odd primes $p$ we determine the fiber functors on $\mathrm{Rep}(H_p)$, where $H_p$ denotes Nikshych's semisimple Hopf algebra of dimension $4p^2$: there is one equivalence class if $p\equiv 3\pmod 4$ and two if $p\equiv 1\pmod 4$. We also use the classification for gaugings to determine which non-pointed entries in the small-dimensional list of Green and Nikshych are representation categories of semisimple factorizable Hopf algebras.

• The paper provides a detailed classification of fiber functors on equivariantizations of finite tensor categories via parameterization by equivariant module categories that satisfy transitivity and non-degeneracy conditions.
• It employs module category theory and explicit cohomology computations to bridge both group-theoretical and non-group-theoretical cases, enhancing Tannaka–Krein duality insights.
• The approach establishes clear criteria for Hopf algebra realizations by highlighting when fiber functors exist or are obstructed through rank and cocycle constraints.

Fiber Functors for Equivariantizations of Finite Tensor Categories

Introduction and Motivation

The problem of classifying fiber functors on finite tensor categories is central for understanding Tannaka--Krein reconstruction and, more generally, for the realization of tensor categories as representation categories of Hopf algebras. For group-theoretical fusion categories, the fiber-functor problem is governed by group-theoretic and cohomological data, but equivariantizations and related constructions yield rich, non-group-theoretical examples, especially in the context of non-symmetric fusion categories.

This paper addresses the explicit classification of fiber functors for equivariantizations $C^G$ of a finite tensor category $C$ under the action of a finite group $G$, without a semisimplicity assumption on $C$. The main result provides a parameterization of fiber functors in terms of equivariant exact module categories, indexed by subgroups of $G$ and satisfying further transitivity and non-degeneracy conditions. Consequently, the results encompass both group-theoretical and non-group-theoretical cases, subsuming classical constructions and producing new structural implications for semisimple and factorizable Hopf algebras.

Classification Theorem for Fiber Functors on Equivariantizations

Let $C$ be a finite tensor category with strict $G$-action, and denote by $C^G$ its equivariantization. The authors establish that the datum of an equivalence class of fiber functor on $C^G$ is equivalent to that of a conjugacy class of pairs $(H, M)$, where $C$0 is a subgroup and $C$1 is an indecomposable, exact, semisimple $C$2-equivariant $C$3-module category. The following sharp rank and cohomological conditions must be met:

• $C$4 acts transitively on the isomorphism classes of simple objects $C$5,
• for some (and hence any) simple $C$6, the stabilizer cocycle $C$7 is non-degenerate.

Under Tannaka--Krein duality, this completely describes all possible Hopf algebra realizations of $C$8. The classification is precise and covers not only the semisimple but also the non-semisimple setting, relying on the correspondence between fiber functors and rank-one exact module categories, together with the theory of module categories over equivariantizations.

In the case where the relevant group cohomology $C$9 vanishes for all subgroups $G$0 (e.g., if all subgroups are cyclic), the conditions reduce to classifying free and transitive actions of $G$1 on $G$2.

Explicit Applications: Hopf Algebras and Gaugings

Fiber Functors for Nikshych’s Semisimple Hopf Algebras

An essential application is to the family $G$3 arising as the equivariantization of Tambara--Yamagami categories $G$4 under an involutive automorphism. The precise statement is as follows: for odd primes $G$5, there is a unique equivalence class of fiber functors if $G$6, and two if $G$7. This distinction is tightly controlled by the existence of $G$8-invariant rank-two module categories and the splitting orbits under the group action. The proof involves careful use of module category theory over the relevant Tambara--Yamagami categories, explicit computation of associated cohomology, and examination of the action on Lagrangian decompositions.

Clifford Theory, Gaugings, and Obstructions

The Clifford-theoretic perspective is used to treat gaugings: given a faithful $G$9-crossed braided extension $C$0 of a braided fusion category $C$1, one obtains a detailed parameterization of the fiber functors on $C$2 in terms of:

• Clifford data $C$3, involving a neutral $C$4-module category $C$5 and its extension $C$6,
• equivariant lifts and transitivity data,
• rank constraints and stabilizer cocycle non-degeneracy, in the same sense as above.

For prime cyclic $C$7 and extensions where cohomological obstructions vanish, the only fiber functors correspond to certain induced module structures and equivariant module categories where the generator acts as a $C$8-cycle on simple objects.

The method is applied to the classification of fiber functors for the small radical gaugings (non-pointed non-degenerate fusion categories with small Frobenius-Perron dimension) cataloged in [GNradical]. Sharp negative results are obtained: for most entries, integrality or structural constraints forbid the existence of fiber functors, and thus of semisimple Hopf (or factorizable) algebra realizations. The only positive cases are ordinary dihedral doubles and specific prime-cyclic twisted doubles associated to hyperbolic forms, with general cohomology class parameter $C$9.

Structure Theory and Cohomological Constraints

The paper carefully elucidates how module categories and their equivariantizations encode subtle invariants—rank, semisimplicity, exactness, and actions on simple objects—providing tools to analyze when equivariantizations can or cannot have Hopf algebra realizations. The stabilizer cocycle condition recovers classical projective representation-theoretic data and characterizes simplicity of the relevant twisted group algebras.

The techniques demonstrate that in many non-pointed and integral cases, even categories with fusion rules compatible with group-theoretical categories lack fiber functors due to explicit cohomological or module-theoretic obstructions. For categories arising as centers or as non-group-theoretical G-crossed extensions, the lack of fiber functors is tightly connected to the absence of free actions or Lagrangian subgroups invariant under the relevant symmetry.

Implications and Prospects

The explicit and highly refined classification results significantly advance the theory of tensor categories, providing an operational framework for recognizing Tannakian or "of Hopf type" categories amongst equivariantizations. In particular, the methods allow one to produce negative results—ruling out the existence of fiber functors in settings where classical group-theoretical intuition might fail—and generate sharp stratifications between group-theoretical, weakly group-theoretical, and non-group-theoretical fusion categories.

Practically, these results facilitate the classification of finite-dimensional semisimple (and factorizable) Hopf algebras by separating those categories admitting fiber functors from those that cannot, informed by the combinatorics of group actions, the representation theory of module categories, and specific group cohomology computations.

Future work can extend these methods to more general crossed or equivariantized categories, adopt higher categorical perspectives (e.g., for 2-representations), and apply these obstructions within the study of topological phases and modular categories, where the existence or non-existence of fiber functors has implications for symmetry-protected phases and quantum group realization questions.

Conclusion

This paper provides a thorough and systematic classification of fiber functors on equivariantizations $G$0 of finite tensor categories under finite group actions, subsuming previous results for group-theoretical categories and extending them to broad non-group-theoretical and non-semisimple settings. The structural results, explicit applications, and comprehensive exclusions derived herein bring powerful new tools to the study of tensor categories, Hopf algebras, and their module-theoretic invariants. The rigid character of the classification facilitates a deeper understanding of the possible fusion and representation-theoretic behaviors that arise in equivariantization constructions.

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Reference: "Fiber Functors of Equivariantizations of Finite Tensor Categories" (2607.10525)