Locally Nilpotent Derivations of Free Algebra of Rank Two

TL;DR

This paper investigates locally nilpotent derivations of the free associative algebra in two variables, showing they are uniquely determined by their kernels, which are free algebras with explicit generators.

Contribution

It extends the understanding of locally nilpotent derivations from commutative polynomial algebras to free associative algebras, characterizing their kernels and uniqueness.

Findings

Derivations are determined up to a constant by their kernels.

Kernels are free associative algebras.

Explicit generators for the kernels are provided.

Abstract

In commutative algebra, if $\delta$ is a locally nilpotent derivation of the polynomial algebra $K[x_1,\ldots,x_d]$ over a field $K$ of characteristic 0 and $w$ is a nonzero element of the kernel of $\delta$, then $\Delta=w\delta$ is also a locally nilpotent derivation with the same kernel as $\delta$. In this paper we prove that the locally nilpotent derivation $\Delta$ of the free associative algebra $K\langle X,Y\rangle$ is determined up to a multiplicative constant by its kernel. We show also that the kernel of $\Delta$ is a free associative algebra and give an explicit set of its free generators.