Picard group of connected affine algebraic group

Abstract

We prove that the Picard group of a connected affine algebraic group $G$ is isomorphic to the fundamental group of the derived subgroup of the reductive algebraic group $G/{\mathcal{R}}_u(G)$, where ${\mathcal{R}}_u(G)$ is the unipotent radical of $G$.

Revised version emphasizing the canonical nature of the construction. 4 pages