The 4--CB Algebra and Solvable Lattice Models

TL;DR

This paper explores the algebraic structures underlying solvable lattice models, proposing a universal algebra depending on the number of blocks, and connects these to knot theory through new link invariants.

Contribution

It introduces a universal algebra framework for IRF lattice models based on the number of blocks and conjectures new algebraic relations for four-block models, supported by evidence.

Findings

Three-block models obey a BMW algebra variant

Four-block models conjectured to follow a modified BMW algebra with additional relations

New link invariants depending on three parameters are proposed for four-block models

Abstract

We study the algebras underlying solvable lattice models of the type fusion interaction round the face (IRF). We propose that the algebras are universal, depending only on the number of blocks, which is the degree of polynomial equation obeyed by the Boltzmann weights. Using the Yang--Baxter equation and the ansatz for the Baxterization of the models, we show that the three blocks models obey a version of Birman--Murakami--Wenzl (BMW) algebra. For four blocks, we conjecture that the algebra is the BMW algebra with a different skein relation, along with one additional relation, and we provide evidence for this conjecture. We connect these algebras to knot theory by conjecturing new link invariants. The link invariants, in the case of four blocks, depend on three arbitrary parameters. We check our result for $G_2$ model with the seven dimensional representation and for $SU(2)$ with the isospin $3/2$ representation, which are both four blocks theories.