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

TL;DR

This paper introduces a new categorical definition of Frobenius structures on algebra objects, applies it to Temperley-Lieb categories at roots of unity, and explores the properties of the Nakayama morphism.

Contribution

It generalizes Frobenius algebras to categorical settings, defines Nakayama morphisms in pivotal categories, and analyzes a specific algebra object in Temperley-Lieb categories.

Findings

The algebra object in Temperley-Lieb category has a Frobenius structure.

The Nakayama morphism of this algebra has order 2.

Results provide insights into Nakayama morphisms of Dynkin preprojective algebras.

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.