On different approaches to IRF lattice models. Part II

A bstract This paper represents a continuation of our previous work, where the Boltzmann weights (BWs) for several Interaction-Round-the Face (IRF) lattice models were computed using their relation to rational conformal field theories. Here, we focus on deriving solutions for the Boltzmann weights of the Interaction-Round the Face lattice model, specifically, the unrestricted face model, based on the $$ \mathfrak{su}{(3)}_k $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>su</mml:mi> <mml:msub> <mml:mfenced> <mml:mn>3</mml:mn> </mml:mfenced> <mml:mi>k</mml:mi> </mml:msub> </mml:math> affine Lie algebra. The admissibility conditions are defined by the adjoint representation. We find the BWs by determining the quantum R matrix of the $$ {U}_q\left(\mathfrak{sl}(3)\right) $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>U</mml:mi> <mml:mi>q</mml:mi> </mml:msub> <mml:mfenced> <mml:mrow> <mml:mi>sl</mml:mi> <mml:mfenced> <mml:mn>3</mml:mn> </mml:mfenced> </mml:mrow> </mml:mfenced> </mml:math> quantum algebra in the adjoint representation and then applying the so-called Vertex-IRF correspondence. The Vertex-IRF correspondence defines the BWs of IRF models in terms of R matrix elements.