Graded cellular bases for Temperley-Lieb algebras of type A and B

TL;DR

This paper demonstrates that the Temperley-Lieb algebra of type A and the blob algebra of type B at roots of unity are both $ extbf{Z}$-graded and graded cellular, enabling their cell modules to be viewed as graded modules.

Contribution

It establishes that these algebras are $ extbf{Z}$-graded and possess graded cellular structures, providing a new framework for understanding their modules.

Findings

Temperley-Lieb algebra of type A is $ extbf{Z}$-graded.

Blob algebra of type B is $ extbf{Z}$-graded.

Both algebras are graded cellular, allowing graded module structures.

Abstract

We show that the Temperley-Lieb algebra of type $A$ and the blob algebra (also known as the Temperley-Lieb algebra of type $ B$) at roots of unity are $ \mathbb Z$-graded algebras.We moreover show that they are graded cellular algebras, thus making their cell modules, or standard modules, graded modules for the algebras.