Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 19 (2023), 075, 42 pages      arXiv:2303.04493      https://doi.org/10.3842/SIGMA.2023.075

Frobenius Monoidal Functors of Dijkgraaf-Witten Categories and Rigid Frobenius Algebras

Samuel Hannah a, Robert Laugwitz b and Ana Ros Camacho a
a) School of Mathematics, Cardiff University, Abacws, Senghennydd Road, Cardiff, CF24 4AG, Wales, UK
b) School of Mathematical Sciences, University of Nottingham, University Park, Nottingham, NG7 2RD, UK

Received March 16, 2023, in final form September 26, 2023; Published online October 12, 2023

Abstract
We construct a separable Frobenius monoidal functor from $\mathcal{Z}\big(\mathsf{Vect}_H^{\omega|_H}\big)$ to $\mathcal{Z}\big(\mathsf{Vect}_G^\omega\big)$ for any subgroup $H$ of $G$ which preserves braiding and ribbon structure. As an application, we classify rigid Frobenius algebras in $\mathcal{Z}\big(\mathsf{Vect}_G^\omega\big)$, recovering the classification of étale algebras in these categories by Davydov-Simmons [J. Algebra 471 (2017), 149-175, arXiv:1603.04650] and generalizing their classification to algebraically closed fields of arbitrary characteristic. Categories of local modules over such algebras are modular tensor categories by results of Kirillov-Ostrik [Adv. Math. 171 (2002), 183-227, arXiv:math.QA/0101219] in the semisimple case and Laugwitz-Walton [Comm. Math. Phys., to appear, arXiv:2202.08644] in the general case.

Key words: Frobenius monoidal functor; Frobenius algebra; Dijkgraaf-Witten category; local module; modular tensor category; étale algebra.

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