Centralizers of Linear and Locally Nilpotent Derivations

Abstract

Let 𝕂 be an algebraically closed field of characteristic zero, let 𝕂[x₁,…,x_n] be the polynomial algebra, and let W_n(𝕂) be the Lie algebra of all 𝕂-derivations on 𝕂[x₁,…,x_n]. For any derivation D with linear components, we describe the centralizer of D in W_n(𝕂) and propose an algorithm for finding the generators of this centralizer regarded as a module over the ring of constants of the derivation D in the case where D is a basic Weitzenböck derivation. In a more general case where a finitely generated integral domain A over the field 𝕂 is considered instead of the polynomial algebra 𝕂[x₁,…,x_n] and D is a locally nilpotent derivation on A, we prove that the centralizer C_DerA(D) of D in the Lie algebra DerA of all 𝕂-derivations on A is a “large” subalgebra of Der A. Specifically, the rank of C_DerA(D) over A is equal to the transcendence degree of the field of fractions Frac(A) over the field 𝕂.