# Integrable models from singly generated planar algebras

## TL;DR

This paper classifies singly generated planar algebras that can encode Yang--Baxter integrable models, identifying three such algebras and demonstrating their Baxterisation and polynomiality properties.

## Contribution

It proves that only three singly generated planar algebras underlie homogeneous Yang--Baxter integrable models and shows the Baxterisation and polynomiality of the associated transfer operators.

## Key findings

- Identifies three singly generated planar algebras related to Yang--Baxter integrability.
- Shows the Liu planar algebra admits a Baxterisation.
- Demonstrates the transfer operator is polynomial in a spectral-parameter-independent element.

## Abstract

Not all planar algebras can encode the algebraic structure of a Yang--Baxter integrable model described in terms of a so-called homogeneous transfer operator. In the family of subfactor planar algebras, we focus on the ones known as singly generated and find that the only such planar algebras underlying homogeneous Yang--Baxter integrable models are the so-called Yang--Baxter relation planar algebras. According to a result of Liu, there are three such planar algebras: the well-known Fuss--Catalan and Birman--Wenzl--Murakami planar algebras, in addition to one more which we refer to as the Liu planar algebra. The Fuss--Catalan and Birman--Wenzl--Murakami algebras are known to underlie Yang--Baxter integrable models, and we show that the Liu algebra likewise admits a Baxterisation. We also show that the homogeneous transfer operator describing a model underlied by a singly generated Yang--Baxter relation planar algebra is polynomialisable, meaning that it is polynomial in a spectral-parameter-independent element of the algebra.