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Standard identities for polynomial vector fields
Abstract
Let Wn be the Lie algebra of polynomial vector fields in n variables over a field of characteristic zero, and let sk denote the alternating sum of compositions of k vector fields. The identity sn2+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 s35, this shows that $35$ is the least degree of a standard identity on W5. We then prove that sn2+2n−1=0 for every even n≥2. 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.
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BibTeX
@misc{shyntas2026standard,
title = {{Standard identities for polynomial vector fields}},
author = {Nurtas Shyntas},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.18149},
eprint = {2610.18149},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.18149},
}
RIS
TY - UNPB
TI - Standard identities for polynomial vector fields
AU - Nurtas Shyntas
PY - 2026
DA - 2026/10/13
PB - arXiv
DO - 10.48550/arxiv.2610.18149
UR - https://arxiv.org/abs/2610.18149
AB - 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≥2. 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.
ER -