Twisted semigroup algebras

We study 2-cocycle twists, or equivalently Zhang twists, of semigroup algebras over a field k. If the underlying semigroup is affine, that is abelian, cancellative and finitely generated, then Spec k[S] is an affine toric variety over k, and we refer to the twists of k[S] as quantum affine toric varieties. We show that every quantum affine toric varieties 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 survive the deformation process.