Frobenius algebra objects in Temperley-Lieb categories at roots of unity

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 Item availability restricted.

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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
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:	08 Dec 2025 10:00
URI:	https://orca.cardiff.ac.uk/id/eprint/176998

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