幂零李代数的几何与代数不变量及其计算
Invariants of Nilpotent Lie Algebras via Geometry and Algebra with a Focus on Computation
- Universidade Federal de Minas Gerais（米纳斯吉拉斯联邦大学）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文比较积分曲线法与Dixmier映射法计算幂零李代数有理不变量，提出核恢复条件并证明三角导子适用，给出生成元算法及SageMath实现。
AI中文摘要：
我们考虑计算幂零李代数有理不变量的问题。我们比较了常用于此任务的两种方法：积分曲线法和Dixmier映射法。给定一个具有多项式系数的有理函数域的导子，我们提出了一个条件，在该条件下，核可以从一族有理积分曲线中恢复，并且我们证明了三角导子满足这一假设。这给出了核作为纯超越扩张的显式描述，并产生了代数独立的生成元。我们还证明了，在三角情形下，所得生成元与通过局部截线从Dixmier映射获得的生成元一致。对由此方法获得的生成集的仔细分析导致了一种计算幂零李代数有理不变量域生成元的算法。这些方法的实现可在SageMath系统中获得。
英文摘要：
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.
