Algebraic embeddings of $\mathbb{C}$ into $\textrm{SL}_n(\mathbb{C})$

TL;DR

This paper demonstrates that for dimensions three and higher, all algebraic embeddings of the complex line into special linear groups are equivalent under algebraic automorphisms, and for dimension two, they are equivalent under holomorphic automorphisms.

Contribution

It establishes the uniqueness of algebraic embeddings of  into extrm{SL}_n(\u001c) up to automorphisms, with a distinction between algebraic and holomorphic cases.

Findings

All algebraic embeddings of  into extrm{SL}_n() are equivalent for n .

Two algebraic embeddings into extrm{SL}_2() are equivalent under holomorphic automorphisms.

The result clarifies the automorphism groups acting on embeddings of  into these groups.

Abstract

We prove that any two algebraic embeddings of $\mathbb{C}$ into $\textrm{SL}_n(\mathbb{C})$ are the same up to an algebraic automorphism of $\textrm{SL}_n(\mathbb{C})$, provided that $n$ is at least $3$. Moreover, we prove that two algebraic embeddings of $\mathbb{C}$ into $\textrm{SL}_2(\mathbb{C})$ are the same up to a holomorphic automorphism of $\textrm{SL}_2(\mathbb{C})$.