# Group actions on algebraic stacks via butterflies

## TL;DR

This paper develops a method to analyze group actions on algebraic stacks, exemplified by weighted projective stacks, and demonstrates how such actions can be lifted to linear actions under certain conditions.

## Contribution

It introduces an explicit approach for studying group stack actions on algebraic stacks and describes automorphism groups of weighted projective stacks in detail.

## Key findings

- Explicit description of automorphism group stacks for weighted projective stacks
- Conditions under which group actions lift to linear actions on affine space
- Application of Colliot-Thelene's result to lift actions for reductive groups

## Abstract

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(n_0,...,n_r) 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, the weighted projective general linear 2-group PGL(n_0,...,n_r). As an application, we use a result of Colliot-Thelene 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(n_0,...,n_r) lifts to a linear action of G on A^{r+1}.