# Modular Valenced Temperley-Lieb Algebras

### Robert A. Spencer

**Abstract:**We investigate the representation theory of the valenced Temperley-Lieb algebras in mixed characteristic. These algebras, as described in characteristic zero by Flores and Peltola, arise naturally in statistical physics and conformal field theory and are a natural deformation of normal Temperley-Lieb algebras. In general characteristic, they encode the fusion rules for the category of \(U_q(\mathfrak{sl}_2)\) tilting modules.

We use the cellular properties of the Temperley-Lieb algebras to determine those of the valenced Temperley-Lieb algebras. Our approach is, at heart, entirely diagrammatic and we calculate cell indices, module dimensions and indecomposable modules for a wide class of valenced Temperley-Lieb algebras. We present a general framework for finding bases of cell modules and a formula for their dimensions.

A draft of this work is available on the ArXiv.

#### Cite with BibTex

```
@misc{spencer2021modular,
title={Modular Valenced Temperley-Lieb Algebras},
author={R. A. Spencer},
year={2021},
eprint={2108.10011},
archivePrefix={arXiv},
primaryClass={math.RT}
}
```