Integrability of planar-algebraic models

The Quantum Inverse Scattering Method is a scheme for solving integrable models in $1+1$ dimensions, building on an $R$-matrix that satisfies the Yang–Baxter equation and in terms of which one constructs a commuting family of transfer matrices. In the standard formulation, this $R$-matrix acts on a tensor product of vector spaces. Here, we relax this tensorial property and develop a framework for describing and analysing integrable models based on planar algebras, allowing non-separable $R\mathit{\text{-operators}}$ satisfying $\mathit{\text{generalised}}$ Yang–Baxter equations. We also re-evaluate the notion of integrals of motion and characterise when an (algebraic) $\mathit{\text{transfer}}\mathit{\text{operator}}$ is polynomial in a single integral of motion. We refer to such models as {\em polynomially integrable}. In an eight-vertex model, we demonstrate that the corresponding transfer operator is polynomial in the natural hamiltonian. In the Temperley–Lieb loop model with loop fugacity $\beta \in \mathbb{C}$, we likewise find that, for all but finitely many $\beta$-values, the transfer operator is polynomial in the usual hamiltonian element of the Temperley–Lieb algebra ${\mathrm{T}\mathrm{L}}_n(\beta )$, at least for $n\le 17$. Moreover, we find that this model admits a second canonical hamiltonian, and that this hamiltonian also acts as a polynomial integrability generator for small $n$ and all but finitely many $\beta$-values.