Group actions on algebraic stacks via butterflies

We introduce an explicit method for studying actions of a group stack G on an algebraic stack X. As an example, we study in detail the case where X=P(n0,⋯,nr) is a weighted projective stack over an arbitrary base S. To this end, we give an explicit description of the group stack of automorphisms of P(n0,⋯,nr), the weighted projective general linear 2-group PGL(n0,⋯,nr). As an application, we use a result of Colliot-Thélène to show that for every linear algebraic group G over an arbitrary base field k (assumed to be reductive if char(k)>0) such that Pic(G)=0, every action of G on P(n0,⋯,nr) lifts to a linear action of G on Ar+1.