The Blob Algebra and the Periodic Temperley-Lieb Algebra

TL;DR

This paper analyzes the structure of the blob algebra and the periodic Temperley-Lieb algebra, providing a comprehensive understanding of their generic and exceptional cases for boundary conditions in statistical mechanics.

Contribution

It introduces the blob algebra and completes the structural analysis of the periodic Temperley-Lieb algebra, building on previous work and connecting the two algebraic structures.

Findings

Determined the generic and exceptional structures of the blob algebra.

Completed the analysis of the periodic Temperley-Lieb algebra.

Connected the structures of both algebras for boundary conditions in models.

Abstract

We determine the structure of two variations on the Temperley-Lieb algebra, both used for dealing with special kinds of boundary conditions in statistical mechanics models. The first is a new algebra, the `blob' algebra (the reason for the name will become obvious shortly!). We determine both the generic and all the exceptional structures for this two parameter algebra. The second is the periodic Temperley-Lieb algebra. The generic structure and part of the exceptional structure of this algebra have already been studied. Here we complete the analysis, using results from the study of the blob algebra.