Locally Nilpotent Derivations of Free Algebra of Rank Two

Abstract

In commutative algebra, if $\delta$ is a locally nilpotent derivation of the polynomial algebra $K[x_1,\dots ,x_d]$ 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 $K⟨X,Y⟩$ 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.

arXiv commentDedicated to the 80th anniversary of Dmitry Fuchs

References in corpus (1)

• Constants of Weitzenböck derivations and invariants of unipotent transformations acting on relatively free algebras