Birational Equivalences for Kac--Moody Borel Enveloping Algebras

A Coxeter ordering of the simple roots of a finite-rank Kac–Moody algebra determines a finite family of commuting real-root vectors. We prove that $U^{\ge 0}(\mathfrak{g})$ is birationally equivalent to $Z\otimes {\mathbb{A}}_n$, where $Z$ is the residual Coxeter centralizer, by identifying the Coxeter localization $U^{\ge 0}(\mathfrak{g})[{\mathbf{X}}^{-1}]$ with $Z\otimes {\mathbb{A}}_n[{\mathbf{x}}^{-1}]$. For symmetrizable Cartan matrices the residual algebra is generated by finite Coxeter windows and is finitely presented. For the generic quantum Borel with torus dual to the root lattice, we prove the analogous birational equivalence.