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Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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Frobenius monoidal functors from (co)Hopf adjunctions
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by Harshit Yadav
Proc. Amer. Math. Soc. 152 (2024), 471-487
DOI: https://doi.org/10.1090/proc/16494
Published electronically: November 17, 2023

Abstract:

Let $U:\mathcal {C}\rightarrow \mathcal {D}$ be a strong monoidal functor between abelian monoidal categories admitting a right adjoint $R$, such that $R$ is exact, faithful and the adjunction $U\dashv R$ is coHopf. Building on the work of Balan [Appl. Categ. Structures 25 (2017), pp. 747–774], we show that $R$ is separable (resp., special) Frobenius monoidal if and only if $R(\mathbb {1}_{\mathcal {D}})$ is a separable (resp., special) Frobenius algebra in $\mathcal {C}$. If further, $\mathcal {C},\mathcal {D}$ are pivotal (resp., ribbon) categories and $U$ is a pivotal (resp., braided pivotal) functor, then $R$ is a pivotal (resp., ribbon) functor if and only if $R(\mathbb {1}_{\mathcal {D}})$ is a symmetric Frobenius algebra in $\mathcal {C}$. As an application, we construct Frobenius monoidal functors going into the Drinfeld center $\mathcal {Z}(\mathcal {C})$, thereby producing Frobenius algebras in it.
References
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Bibliographic Information
  • Harshit Yadav
  • Affiliation: Department of Mathematics, University of Alberta, Edmonton, Alberta, Canada T6G 2G1
  • MR Author ID: 1428044
  • ORCID: 0009-0006-4623-4998
  • Email: hyadav3@ualberta.ca
  • Received by editor(s): October 31, 2022
  • Received by editor(s) in revised form: March 7, 2023, and March 25, 2023
  • Published electronically: November 17, 2023
  • Additional Notes: The author was partially supported by Nettie S. Autrie Research Fellowship from Rice University.
  • Communicated by: Sarah Witherspoon
  • © Copyright 2023 by Harshit Yadav
  • Journal: Proc. Amer. Math. Soc. 152 (2024), 471-487
  • MSC (2020): Primary 16T05, 18M15, 18M20
  • DOI: https://doi.org/10.1090/proc/16494
  • MathSciNet review: 4683832