Twisted Semigroup Algebras

We study 2-cocycle twists, or equivalently Zhang twists, of semigroup algebras over a field K\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}${\mathbb K}$\end{document}. If the underlying semigroup is affine, that is abelian, cancellative and finitely generated, then SpecK[S]\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathsf {Spec}~{\mathbb K}[S]$\end{document} is an affine toric variety over K\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}${\mathbb K}$\end{document}, and we refer to the twists of K[S]\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}${\mathbb K}[S]$\end{document} as quantum affine toric varieties. We show that every quantum affine toric variety has a “dense quantum torus”, in the sense that it has a localization isomorphic to a quantum torus. We study quantum affine toric varieties and show that many geometric regularity properties of the original toric variety survive the deformation process.