Grant, Joseph and Pugh, Mathew  ORCID: https://orcid.org/0000-0001-9045-3713
2025.
Frobenius algebra objects in Temperley-Lieb categories at roots of unity.
Quantum Topology
10.1017/S0017089525100876
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Abstract
We give a new definition of a Frobenius structure on an algebra object in a monoidal category, generalising Frobenius algebras in the category of vector spaces. Our definition allows Frobenius forms valued in objects other than the unit object, and can be seen as a categorical version of Frobenius extensions of the second kind. When the monoidal category is pivotal we define a Nakayama morphism for the Frobenius structure and explain what it means for this morphism to have finite order. Our main example is a well-studied algebra object in the (additive and idempotent completion of the) Temperley-Lieb category at a root of unity. We show that this algebra has a Frobenius structure and that its Nakayama morphism has order 2. As a consequence, we obtain information about Nakayama morphisms of preprojective algebras of Dynkin type, considered as algebras over the semisimple algebras on their vertices.
| Item Type: | Article |
|---|---|
| Date Type: | Published Online |
| Status: | In Press |
| Schools: | Schools > Mathematics |
| Subjects: | Q Science > QA Mathematics |
| Publisher: | EMS Press |
| ISSN: | 1663-487X |
| Date of First Compliant Deposit: | 27 November 2025 |
| Date of Acceptance: | 27 November 2025 |
| Last Modified: | 27 Apr 2026 09:53 |
| URI: | https://orca.cardiff.ac.uk/id/eprint/176998 |
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