Hyperbolicity of Semigroup Algebras

Abstract

Let $A$ be a finite dimensional $Q-$algebra and $\Gamma subsetA$ a $Z-$order. We classify those $A$ with the property that $Z^2$ does not embed in $\mathcal{U}(\Gamma )$. We call this last property the hyperbolic property. We apply this in the case that $A=KS$ a semigroup algebra with $K=Q$ or $K=Q(\sqrt{-d})$. In particular, when $KS$ is semi-simple and has no nilpotent elements, we prove that $S$ is an inverse semigroup which is the disjoint union of Higman groups and at most one cyclic group $C_n$ with $n\in \{5,8,12\}$.

This article corresponds to the second chapter of the third author PhD Thesis