Binomial Artin-Schelter Regular Algebras and Twisted Monoid Algebras

We study four-dimensional binomial Artin-Schelter regular algebras and their connection with twisted monoid algebras. Starting with $23$ examples up to permutation of generators, we investigate the associated monoids of I-type and the structure of their groups of fractions. We determine the corresponding finite permutation groups and compute the second group cohomology $H^2(G(M),k^\times)$. These computations allow us to construct explicit families of four-dimensional Koszul Artin-Schelter regular algebras through cocycle twists. We illustrate this construction for some of these algebras.