Picture groups and green sequences from the perspective of Ringel--Hall algebras

Abstract

Let $k$ be an algebraically closed field and let $Λ$ be a finite-dimensional associative $k$-algebra. We apply Joyce and Brideland's notion of Ringel--Hall algebra to prove results about picture spaces and picture groups. For instance, we construct a faithful group functor for $Λ$, i.e. a faithful functor from the $τ$-cluster morphism category of $Λ$ into a groupoid. Consequently, by results of Hanson--Igusa, the picture space of $Λ$ is locally CAT(0) provided that the $τ$-cluster morphism category of $Λ$ admits compatibility of last factors. We also show that green sequences are in bijection with certain positive expressions in the picture group, which generalizes a result of Igusa--Todorov beyond hereditary $k$-algebras.