Uniqueness of Embeddings of the Affine Line into Algebraic Groups

TL;DR

This paper proves that embeddings of the affine line into certain algebraic groups are unique up to automorphism, except for specific product varieties involving tori and low-dimensional groups.

Contribution

It establishes the uniqueness of affine line embeddings into a broad class of algebraic groups, excluding some well-understood product cases.

Findings

Embeddings are unique up to automorphism for most affine algebraic groups.

Exceptions occur only when the variety is a product involving a torus and specific low-dimensional groups.

Provides a classification of when affine line embeddings are not unique.

Abstract

Let $Y$ be the underlying variety of a connected affine algebraic group. We prove that two embeddings of the affine line $\mathbb{C}$ into $Y$ are the same up to an automorphism of $Y$ provided that $Y$ is not isomorphic to a product of a torus $(\mathbb{C}^\ast)^k$ and one of the three varieties $\mathbb{C}^3$, $\operatorname{SL}_2$, and $\operatorname{PSL}_2$.