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Journal Article

Locally nilpotent derivations of free algebra of rank two


Makar-Limanov,  Leonid
Max Planck Institute for Mathematics, Max Planck Society;

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Drensky, V., & Makar-Limanov, L. (2019). Locally nilpotent derivations of free algebra of rank two. Symmetry, Integrability and Geometry: Methods and Applications, 15: 091. doi:10.3842/SIGMA.2019.091.

Cite as: https://hdl.handle.net/21.11116/0000-0008-A80A-0
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.