Locally Nilpotent Derivations of Free Algebra of Rank Two

Abstract

In commutative algebra, if δ is a locally nilpotent derivation of the polynomial algebra 𝛫[x₁, …, xd] over a field 𝛫 of characteristic 0 and w is a nonzero element of the kernel of δ, then Δ=wδ is also a locally nilpotent derivation with the same kernel as δ. In this paper, we prove that the locally nilpotent derivation Δ of the free associative algebra 𝛫⟨X, Y⟩ is determined up to a multiplicative constant by its kernel. We also show that the kernel of Δ is a free associative algebra and give an explicit set of its free generators.