Twisted semigroup algebras

TL;DR

This paper investigates 2-cocycle twists of semigroup algebras over fields, focusing on affine semigroups that produce quantum affine toric varieties, and demonstrates that many geometric properties are preserved under these twists.

Contribution

It introduces the concept of quantum affine toric varieties via 2-cocycle twists and shows they contain a dense quantum torus, preserving several geometric regularity properties.

Findings

Quantum affine toric varieties have a dense quantum torus.

Many geometric regularity properties are preserved under twists.

Every quantum affine toric variety contains a localization isomorphic to a quantum torus.

Abstract

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.