Casimir elements and kernel of Weitzenbök derivation

Abstract

Let $k[X]{:=} k[x_0,x_1,..., x_n]$ be a polynomial algebra over a field $k$ of characteristic zero. We offer an algorithm for calculation of kernel of Weitzenbök derivation ${d(x_i)=x_{i-1}}, ...,$ ${d(x_0)=0}$, ${i=1... n}$ that is based on an analogue of the well known Casimir elements of finite dimensional Lie algebras. By using this algorithm, the kernel is calculated in the case $n <7$.