发表机构
University of Illinois at Urbana-Champaign（伊利诺伊大学厄巴纳-香槟分校）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该研究证明特征零域上有限生成交换代数的局部有限导子生成的有限生成李子代数必有限维，解决了Popov2005年关于幂单单参数子群的问题，还刻画了多项式控制系统的双线性实现。
AI 中文摘要
设A是特征零域𝕂上有限生成交换代数。我们证明，每个满足元素在A上局部有限的有限生成李子代数L⊆Der_𝕂(A)都是有限维的。由此，对于由有限个局部有限导子生成的李子代数，以下三者等价：它是有限维的，它在A上局部有限作用，且其所有元素都是局部有限的。一个关键组成部分是另一个具有独立意义的定理：Der_𝕂(A)中所有元素局部幂零的李子代数都是可解的；当A既约时，其导出长度至多为dim A，且该界是紧的。对于代数闭域上的仿射簇X，我们由此推出：由有限个连通代数子群生成的Aut(X)子群是代数的，当且仅当由它们的切代数生成的李代数的每个元素都是局部有限的。对于两个幂单单参数子群，这为解决Popov在2005年提出的问题提供了答案。我们的结果还刻画了具有精确有限维双线性实现的多项式控制系统，该实现包含状态坐标的多项式可观测量。证明依赖于引入一个余有限理想，该理想与Spec A的每个伴随点的闭包相交：在沿对应有限子概型二阶消失的导子子代数上，局部有限性迫使局部幂零性。因此，该子代数与L的交由第二个定理可知是可解的，且在L中具有有限余维数；结合伴随作用的局部有限性，即可得到有限维性。
英文摘要
Let $A$ be a finitely generated commutative algebra over a field $\mathbb K$ of characteristic zero. We prove that every finitely generated Lie subalgebra of $\operatorname{Der}_{\mathbb K}(A)$ whose elements are locally finite on $A$ is finite-dimensional. Consequently, for a Lie subalgebra generated by finitely many locally finite derivations, the following are equivalent: it is finite-dimensional, it acts locally finitely on $A$, and all its elements are locally finite. A key ingredient is a second theorem: every Lie subalgebra of $\operatorname{Der}_{\mathbb K}(A)$ whose elements are locally nilpotent is solvable; when $A$ is reduced, its derived length is at most $\dim A$, and this bound is sharp. In general, the $(\dim A)$-th term of its derived series is nilpotent, of class bounded in terms of $A$ only. We also prove that every solvable Lie subalgebra of $\operatorname{Der}_{\mathbb K}(A)$ generated by finitely many locally finite derivations is finite-dimensional and acts locally finitely on $A$. For an affine variety $X$ over an algebraically closed field, we deduce that a subgroup of $\operatorname{Aut}(X)$ generated by finitely many connected algebraic subgroups is algebraic if and only if every element of the Lie algebra generated by their tangent algebras is locally finite. For two unipotent one-parameter subgroups, this gives an answer to a 2005 problem of Popov. Our results also characterize polynomial control systems admitting an exact finite-dimensional bilinear realization by polynomial observables containing the state coordinates. The proof uses derivations vanishing to second order along a suitable finite subscheme, on which local finiteness forces local nilpotence.
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