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#### Locally nilpotent derivations of free algebra of rank two

##### External Resource

https://doi.org/10.3842/SIGMA.2019.091

(Publisher version)

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Drensky-Makar-Limanov_Locally nilpotent derivations of free algebra of rank two_2019.pdf

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##### Citation

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

##### 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.

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.