Abstract
We study 2-cocycle twists, or equivalently Zhang twists, of semigroup algebras over a field ${\mathbb K}$ . If the underlying semigroup is affine, that is abelian, cancellative and finitely generated, then $\mathsf {Spec}~{\mathbb K}[S]$ is an affine toric variety over ${\mathbb K}$ , and we refer to the twists of ${\mathbb K}[S]$ 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.
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