Invariants of Nilpotent Lie Algebras via Geometry and Algebra with a Focus on Computation
Abstract
We consider the problem of computing rational invariants of nilpotent Lie algebras. We compare two methods that are commonly used for this task: the method of integral curves and the Dixmier map. Given a derivation of a rational function field with polynomial coefficients, we formulate a condition under which the kernel can be recovered from a family of rational integral curves, and we show that triangular derivations satisfy this hypothesis. This yields an explicit description of the kernel as a purely transcendental extension and produces algebraically independent generators. We also show that, in the triangular case, the resulting generators agree with those obtained from the Dixmier map via a local slice. A careful analysis of the generating set obtained from this method leads to an algorithm for computing generators of the rational invariant field of a nilpotent Lie algebra. An implementation of the methods is available in the SageMath system.
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BibTeX
@misc{schneider2026invariants,
title = {{Invariants of Nilpotent Lie Algebras via Geometry and Algebra with a Focus on Computation}},
author = {Csaba Schneider and Igor Martins Silva},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.10956},
eprint = {2609.10956},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.10956},
note = {CC BY-NC-SA 4.0},
}
RIS
TY - UNPB
TI - Invariants of Nilpotent Lie Algebras via Geometry and Algebra with a Focus on Computation
AU - Csaba Schneider
AU - Igor Martins Silva
PY - 2026
DA - 2026/09/10
PB - arXiv
DO - 10.48550/arxiv.2609.10956
UR - https://arxiv.org/abs/2609.10956
AB - We consider the problem of computing rational invariants of nilpotent Lie algebras. We compare two methods that are commonly used for this task: the method of integral curves and the Dixmier map. Given a derivation of a rational function field with polynomial coefficients, we formulate a condition under which the kernel can be recovered from a family of rational integral curves, and we show that triangular derivations satisfy this hypothesis. This yields an explicit description of the kernel as a purely transcendental extension and produces algebraically independent generators. We also show that, in the triangular case, the resulting generators agree with those obtained from the Dixmier map via a local slice. A careful analysis of the generating set obtained from this method leads to an algorithm for computing generators of the rational invariant field of a nilpotent Lie algebra. An implementation of the methods is available in the SageMath system.
ER -