# Classifying Module Categories for Generalized Temperley-Lieb-Jones *-2-Categories

## TL;DR

This paper introduces a classification of unitary modules for generalized Temperley-Lieb-Jones 2-categories associated with weighted bidirected graphs, extending previous classifications in quantum algebra.

## Contribution

It provides a new classification framework for unitary modules of generalized TLJ 2-categories using weighted bi-directed fair and balanced graphs.

## Key findings

- Classified unitary modules up to *-equivalence.
- Extended Yamagami's classification to generalized TLJ categories.
- Connected module classification to weighted bi-directed graphs.

## Abstract

Generalized Temperley-Lieb-Jones (TLJ) 2-categories associated to weighted bidirected graphs were introduced in unpublished work of Morrison and Walker. We introduce unitary modules for these generalized TLJ 2-categories as strong *-pseudofunctors into the *-2-category of row-finite separable bigraded Hilbert spaces. We classify these modules up to *-equivalence in terms of weighted bi-directed fair and balanced graphs in the spirit of Yamagami's classification of fiber functors on TLJ categories and DeCommer and Yamashita's classification of unitary modules for Rep(SUq(2)).