The 5-CB Algebra and Fused $SU(2)$ Lattice Models

We study the fused $SU(2)$ models put forward by Date et al., which is a series of models with arbitrary number of blocks, which is the degree of the polynomial equation obeyed by the Boltzmann weights. We demonstrate by a direct calculation that a version of BMW (Birman--Murakami--Wenzl) algebra is obeyed by five, six and seven blocks models, establishing that it is obeyed for any model with more than two blocks. Previously, we described the algebra for two, three and four blocks. We use the Yang--Baxter equation to describe explicitly the algebra for five blocks, obtaining $19$ additional non--trivial relations. We call this algebra 5--CB (Conformal Braiding) algebra. Our method can be used to describe the algebra for any solvable model of this type and for any number of blocks, limited only by the complexity of the calculation. Our results are of use in the realms of quantum algebras and knot theory.