Three Blocks Solvable Lattice Models and Birman--Murakami--Wenzl Algebra

TL;DR

This paper demonstrates that three block IRF lattice models obey the BMW algebra, linking solvable lattice models with knot theory and conformal field theory, and providing explicit examples with $SU(2)$ models.

Contribution

It proves that three block IRF lattice models satisfy the BMW algebra, expanding the algebraic understanding of these models and their connection to knot theory.

Findings

Three block IRF models obey BMW algebra

Explicit BMW structure in $SU(2)$ fused models

Connection established between lattice models and conformal field theory

Abstract

Birman--Murakami--Wenzl (BMW) algebra was introduced in connection with knot theory. We treat here interaction round the face solvable (IRF) lattice models. We assume that the face transfer matrix obeys a cubic polynomial equation, which is called the three block case. We prove that the three block theories all obey the BMW algebra. We exemplify this result by treating in detail the $SU(2)$ $2\times 2$ fused models, and showing explicitly the BMW structure. We use the connection between the construction of solvable lattice models and conformal field theory. This result is important to the solution of IRF lattice models and the development of new models, as well as to knot theory.