Standard identities for polynomial vector fields

Abstract

Let $W_n$ be the Lie algebra of polynomial vector fields in $n$ variables over a field of characteristic zero, and let $s_k$ denote the alternating sum of compositions of $k$ vector fields. The identity $s_{n^2+2n-1}=0$ has been asserted for arbitrary $n$. We give an explicit counterexample in dimension five, verified by an exact integer computation of a $34$-commutator. Together with the known vanishing of $s_{35}$, this shows that $35$ is the least degree of a standard identity on $W_5$. We then prove that $s_{n^2+2n-1}=0$ for every even $n\geq2$. The proof uses Lie algebra homology with coefficients in the polynomial algebra modulo constants and a lemma on invariant tensors. We describe the computation and provide the source code and verification results.