The Six-Vertex Yang-Baxter Groupoid
Daniel Bump, Slava Naprienko
TL;DR
The paper expands the Yang–Baxter framework from groups to groupoids, enabling a richer parameter space for R-matrices in solvable lattice models. It constructs a six-vertex groupoid consisting of a free-fermionic part and a non-free-fermionic part via a blow-up of boundary components, and also develops a separate five-vertex groupoid, with explicit maps and composition rules tied to an object map $\Delta$. It provides explicit YBE conditions and normalized solutions in both free-fermionic and non-free-fermionic regimes, and demonstrates how to build solvable lattice models with row and column solvability from these groupoid parametrizations. The results extend known FF results, reveal new groupoid structures beyond disjoint unions of groups, and lay groundwork for deeper connections to quantum groups and Hopf algebroids while enabling broadening of solvable model constructions.
Abstract
A parametrized Yang-Baxter equation is usually defined to be a map from a group to a set of R-matrices, satisfying the Yang-Baxter commutation relation. These are a mainstay of solvable lattice models. We will show how the parameter space can sometimes be enlarged to a groupoid, and give two examples of such groupoid parametrized Yang-Baxter equations, within the six vertex model. A groupoid parametrized Yang-Baxter equation consists of a groupoid $\mathfrak{G}$ together with a map $π:\mathfrak{G}\to\operatorname{End}(V\otimes V)$ for some vector space $V$ such that the Yang-Baxter commutator $[[ π(u),π(w),π(v)]]=0$ if $u,v\in\mathfrak{G}$ are such that the groupoid composition $w=u\star v$ is defined. An important role is played by an object map $Δ:\mathfrak{G}\to M$ for some set $M$ such that $Δ(u)=Δ(v')$, $Δ(w)=Δ(v)$ and $Δ(w')=Δ(u')$, where $v\mapsto v'$ is the groupoid inverse map. There are two main regimes of the six-vertex model: the free-fermionic point, and everything else. For the free-fermionic point, there exists a parametrized Yang-Baxter equation with a large parameter group $\operatorname{GL}(2)\times\operatorname{GL}(1)$. For non-free-fermionic six-vertex matrices, there are also well-known (group) parametrized Yang-Baxter equations, but these do not account for all possible interactions. Instead we will construct a groupoid parametrized Yang-Baxter equation that accounts for essentially all possible Yang-Baxter equations in the six-vertex model. We will also exhibit a separate groupoid for the five-vertex model. We will show how to construct solvable lattice models based on groupoid parametrized Yang-Baxter equations.