Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 15 (2019), 091, 10 pages      arXiv:1909.13262      https://doi.org/10.3842/SIGMA.2019.091
Contribution to the Special Issue on Algebra, Topology, and Dynamics in Interaction in honor of Dmitry Fuchs

Locally Nilpotent Derivations of Free Algebra of Rank Two

Vesselin Drensky a and Leonid Makar-Limanov bc
a) Institute of Mathematics and Informatics, Bulgarian Academy of Sciences, 1113 Sofia, Bulgaria
b) Department of Mathematics, Wayne State University Detroit, MI 48202, USA
c) Department of Mathematics, The Weizmann Institute of Science, Rehovot 7610001, Israel

Received October 01, 2019, in final form November 13, 2019; Published online November 18, 2019

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.

Key words: free associative algebras; locally nilpotent derivations; algebras of constants.

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