Centralizers of elements in Lie algebras of vector fields with polynomial coefficients

Abstract

Let K be an algebraically closed field of characteristic zero, A = K[x₁,...,x_n] the polynomial ring, and R = K(x₁,...,x_n) the field of rational functions in n variables. Denote by W_n = W_n(K) the Lie algebra of all K-derivations on A (in case C it is the Lie algebra of all vector fields on Cⁿ with polynomial coefficients). For a given D ∈ W_n(K) the structure of the centralizer C_{Wn (K)}(D) depends on the field of constants kerD = {ϕ ∈ R | D(ϕ)=0}, (here we extend naturally every derivation D of A on the field R). The case tr.deg_K kerD ≤ 1 is studied, the structure of the subalgera C_Wn(K)(D) is characterized, in particular it is proved that if kerD does not contain any non-constant polynomial, then C_Wn(K)(D) is finite-dimensional over K. Some results about centralizers of linear derivations in W_n(K) are obtained.