# Picard group of connected affine algebraic group

## TL;DR

This paper establishes a precise relationship between the Picard group of a connected affine algebraic group and the fundamental group of the derived subgroup of its reductive quotient, linking algebraic and topological invariants.

## Contribution

It proves an isomorphism between the Picard group of a connected affine algebraic group and the fundamental group of a related reductive subgroup, clarifying their structural connection.

## Key findings

- Picard group is isomorphic to the fundamental group of the derived subgroup
- Provides a new link between algebraic and topological invariants of algebraic groups
- Clarifies the structure of the Picard group for connected affine algebraic groups

## 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 $G/{\mathscr R}_u(G)$, where ${\mathscr R}_u(G)$ is the unipotent radical of $G$.