标题： 六顶点杨-巴克斯特群oid 显示英文标题

标题： The Six-Vertex Yang-Baxter Groupoid

摘要： 参数化的Yang-Baxter方程通常定义为从一个群到R-矩阵集合的映射，满足Yang-Baxter交换关系。 这些是可解格点模型中的常见内容。 我们将展示如何有时可以将参数空间扩展为一个群oid，并在六顶点模型中给出两个这样的群oid参数化Yang-Baxter方程的例子。 一个群oid参数化的Yang-Baxter方程由一个群oid$\mathfrak{G}$以及一个映射$\pi:\mathfrak{G}\to\operatorname{End}(V\otimes V)$对某个向量空间$V$，使得Yang-Baxter换位子$[[ \pi(u),\pi(w),\pi(v)]]=0$如果$u,v\in\mathfrak{G}$是使得群oid乘法$w=u\star v$定义的。 一个重要的角色是由对象映射$\Delta:\mathfrak{G}\to M$对于某些集合$M$使得$\Delta(u)=\Delta(v')$,$\Delta(w)=\Delta(v)$和$\Delta(w')=\Delta(u')$，其中$v\mapsto v'$是该叠数的逆映射。 六顶点模型有两种主要情况：自由费米子点，以及其余所有情况。 对于自由费米子点，存在一个带有大参数群的参数化的杨-巴克斯特方程$\operatorname{GL}(2)\times\operatorname{GL}(1)$。 对于非自由费米子的六顶点矩阵，也存在众所周知的（群）参数化的杨-巴克斯特方程，但这些并不能涵盖所有可能的相互作用。 相反，我们将构建一个叠数参数化的杨-巴克斯特方程，它几乎涵盖了六顶点模型中所有可能的杨-巴克斯特方程。 我们还将展示一个独立的叠数用于五顶点模型。 我们将说明如何基于叠数参数化的杨-巴克斯特方程来构建可解的格点模型。 显示英文摘要

摘要： 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 $\pi:\mathfrak{G}\to\operatorname{End}(V\otimes V)$ for some vector space $V$ such that the Yang-Baxter commutator $[[ \pi(u),\pi(w),\pi(v)]]=0$ 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 $\Delta:\mathfrak{G}\to M$ for some set $M$ such that $\Delta(u)=\Delta(v')$, $\Delta(w)=\Delta(v)$ and $\Delta(w')=\Delta(u')$, 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.