Classification of twisted generalized Weyl algebras over polynomial rings

Let $R$ be a polynomial ring in $m$ variables over a field of characteristic zero. We classify all rank $n$ twisted generalized Weyl algebras over $R$, up to ${\mathbb{Z}}^n$-graded isomorphisms, in terms of higher spin 6-vertex configurations. Examples of such algebras include infinite-dimensional primitive quotients of $U(\mathfrak{g})$ where $\mathfrak{g}={\mathfrak{g}\mathfrak{l}}_n$, ${\mathfrak{s}\mathfrak{l}}_n$, or ${\mathfrak{s}\mathfrak{p}}_{2n}$, algebras related to $U({\widehat{\mathfrak{s}\mathfrak{l}}}_2)$ and a finite W-algebra associated to ${\mathfrak{s}\mathfrak{l}}_4$. To accomplish this classification we first show that the problem is equivalent to classifying solutions to the binary and ternary consistency equations. Secondly, we show that the latter problem can be reduced to the case $n=2$, which can be solved using methods from previous work by the authors. As a consequence we obtain the surprising fact that (in the setting of the present paper) the ternary consistency relation follows from the binary consistency relation.