Locally nilpotent derivations and automorphisms of free associative algebra with two generators

Abstract

We prove that every locally nilpotent derivation D of the free associative algebra $A=F〈x,y〉$ over a field of characteristic 0 is triangulable, that is, admits a system of generators $x\prime ,y\prime$ of A such that $D(x\prime )=f(y\prime ),\hspace{0.17em}D(y\prime )=0.$ This is an analog of the well-known Rentschler theorem for the algebra of polynomials $F[x,y].$ As a corollary, we obtain a new proof of the classical Czerniakiewicz–Makar Limanov theorem on the isomorphism of the groups $Aut(F〈x,y〉)$ and $Aut(F[x,y])$ for the case of characteristic 0.

Keywords:

• Locally nilpotent derivation
• free associative algebra
• automorphism

MATHEMATICS SUBJECT CLASSIFICATION:

• Primary: 16S10
• 16W20
• 16W25
• 14R10
• Secondary: 13F20
• 13B25
• 13N15

Acknowledgments

The paper is a part of a PhD thesis of the first author realized at the Institute of Mathematics and Statistics of the University of São Paulo. The authors thank the referee for useful remarks and suggestions.

Additional information

Funding

The first author was supported by the CNPq scholarship. The second author was partially supported by the CNPq grant 304313/2019-0 and FAPESP grant 2018/23690-6.