The ghost algebra and the dilute ghost algebra

TL;DR

This paper introduces the ghost algebra and dilute ghost algebra as new two-boundary generalizations of the Temperley-Lieb algebra, incorporating boundary decorations to allow odd boundary connections, and explores their associated integrable loop models.

Contribution

It presents novel algebraic structures with diagrammatic bases, extends the Temperley-Lieb framework, and classifies boundary Yang-Baxter solutions for these models.

Findings

Defined ghost and dilute ghost algebras with diagrammatic presentations

Classified solutions to boundary Yang-Baxter equations for these algebras

Constructed integrable loop models with commuting transfer tangles

Abstract

We introduce the ghost algebra, a two-boundary generalisation of the Temperley-Lieb (TL) algebra, using a diagrammatic presentation. The existing two-boundary TL algebra has a basis of string diagrams with two boundaries, and the number of strings connected to each boundary must be even; in the ghost algebra, this number may be odd. To preserve associativity while allowing boundary-to-boundary strings to have distinct parameters according to the parity of their endpoints, as seen in the one-boundary TL algebra, we decorate the boundaries with bookkeeping dots called ghosts. We also introduce the dilute ghost algebra, an analogous two-boundary generalisation of the dilute TL algebra. We then present loop models associated with these algebras, and classify solutions to their boundary Yang-Baxter equations, given existing solutions to the Yang-Baxter equations for the TL and dilute TL models. This facilitates the construction of a one-parameter family of commuting transfer tangles, making these models Yang-Baxter integrable.