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

Joseph Grant, Mathew Pugh

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.

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

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.
Paper Structure (11 sections, 28 theorems, 38 equations)

This paper contains 11 sections, 28 theorems, 38 equations.

Table of Contents

  1. Introduction
  2. Frobenius algebras
  3. Frobenius structures
  4. Nakayama automorphisms
  5. Transfer of structure
  6. Examples
  7. The Temperley-Lieb category
  8. Jones-Wenzl projections
  9. Pivotal and braided structures
  10. Twisted Frobenius structure
  11. Preprojective algebras

Key Result

Theorem 1

The algebra object $\Sigma\in\widetilde{TL}$ is Frobenius with Nakayama morphism of order $2$.

Theorems & Definitions (65)

  • Theorem : Lemma \ref{['lem:sigma-frob']} and Theorem \ref{['thm:alpha^2']}
  • Definition 2.1
  • Remark 2.2
  • Proposition 2.3
  • proof
  • Lemma 2.4
  • proof
  • Lemma 2.5
  • proof
  • Definition 2.6
  • ...and 55 more