On the Algebraic Approach to Solvable Lattice Models

TL;DR

This paper explores the algebraic structures of IRF solvable lattice models, establishing the BMW algebra as fundamental for models with three or more blocks and deriving new knot invariants from these algebraic relations.

Contribution

It demonstrates that the BMW algebra underpins IRF models with three or more blocks and introduces new knot invariants based on these algebraic structures.

Findings

Yang-Baxter equation is obeyed iff BMW algebra is obeyed for three blocks

BMW algebra is also obeyed for four blocks, with a different skein relation

New knot invariant depending on three parameters is derived from the algebraic relations

Abstract

We treat here interaction round the face (IRF) solvable lattice models. We study the algebraic structures underlining such models. For the three block case, we show that the Yang Baxter equation is obeyed, if and only if, the Birman--Murakami--Wenzl (BMW) algebra is obeyed. We prove this by an algebraic expansion of the Yang Baxter equation (YBE). For four blocks IRF models, we show that the BMW algebra is also obeyed, apart from the skein relation, which is different. This indicates that the BMW algebra is a sub--algebra for all models with three or more blocks. We find additional relations for the four block algebra using the expansion of the YBE. The four blocks result, that is the BMW algebra and the four blocks skein relation, is enough to define new knot invariant, which depends on three arbitrary parameters, important in knot theory.