Action of derivations on polynomials and on Jacobian derivations

Abstract

Let $$$\mathbb K$$$ be a field of characteristic zero, $$$A := \mathbb K[x_{1}, x_{2}]$$$ the polynomial ring and $$$W_2(\mathbb K)$$$ the Lie algebra of all $$$\mathbb K$$$-derivations on $$$A$$$. Every polynomial $$$f \in A$$$ defines a Jacobian derivation $$$D_f\in W_2(\mathbb K)$$$ by the rule $$$D_f(h)=\det J(f, h)$$$ for any $$$h\in A$$$, where $$$J(f, h)$$$ is the Jacobi matrix for $$$f, h$$$. The Lie algebra $$$W_2(\mathbb K)$$$ acts naturally on $$$A$$$ and on itself (by multiplication). We study relations between such actions from the viewpoint of Darboux polynomials of derivations from $$$W_2(\mathbb K)$$$. It is proved that for a Jordan chain $$$T(f_1)=\lambda f_1+f_2$$$, ..., $$$T(f_{k-1})=\lambda f_{k-1}+f_k$$$, $$$T(f_k)=\lambda f_k$$$ for a derivation $$$T\in W_2(\mathbb K)$$$ on $$$A$$$ there exists an analogous chain $$$[T,D_{f_1}]=(\lambda -\mathop{\mathrm{div}} T)D_{f_1} + D_{f_2}$$$, ..., $$$[T,D_{f_{k}}]=(\lambda -\mathop{\mathrm{div}} T)D_{f_{k}}$$$ in $$$W_2(\mathbb K)$$$. In case $$$A:=\mathbb K[x_1, \ldots , x_n]$$$, the action of normalizers of elements $$$f$$$ from $$$A$$$ in $$$W_n(\mathbb K)$$$ on the principal ideals $$$(f)$$$ is considered.