A Cyclic Approach to the Annular Temperley-Lieb Category

TL;DR

This paper provides an abstract presentation of the annular Temperley-Lieb category, highlighting how two cyclic categories generate it and exploring modules and homologies from a cyclic perspective.

Contribution

It introduces a new abstract framework for ATL, detailing the generation by cyclic categories and analyzing modules and homologies within this structure.

Findings

Two copies of the cyclic category generate ATL.

Modules over ATL are characterized via cyclic structures.

Homologies of modules are derived from the cyclic viewpoint.

Abstract

In 2000, Jones found two copies of the cyclic category in the annular Temperley-Lieb category ATL. We give an abstract presentation of ATL to discuss how these two copies of the cyclic category generate ATL together with the coupling constants and the coupling relations. We then discuss modules over the annular category and homologies of such modules, the latter of which arises from the cyclic viewpoint.