Integrable models from singly generated planar algebras

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.

arXiv comment38 pages, v2: title changed, presentation restructured

References in corpus (10)

• Logarithmic Minimal Models
• The Blob Algebra and the Periodic Temperley-Lieb Algebra
• Interaction-Round-a-Face Models with Fixed Boundary Conditions: The ABF Fusion Hierarchy
• The Annular Structure of Subfactors
• Exact S-matrices for supersymmetric sigma models and the Potts model
• Combinatorial point for higher spin loop models
• Logarithmic Superconformal Minimal Models
• Fusion hierarchies, T-systems and Y-systems of logarithmic minimal models
• New Integrable Lattice Models From Fuss-Catalan Algebras
• Integrability of planar-algebraic models

Cited by in corpus (1)

• The planar parafermion algebra: The ${\mathbb{Z}}_N$ clock model and the coupled Temperley-Lieb algebra