向量场的可递幂零李代数及其田中（Tanaka）延拓
Transitive nilpotent Lie algebras of vector fields and their Tanaka prolongations
AI总结：
本文研究向量场可递幂零李代数的田中延拓导子来源问题，明确一般情形下0次导子由0次向量场给出、延拓恢复整个多项式向量代数，同时找出并刻画了特定伸缩情形下的奇异导子。
摘要：
向量场的可递局部李代数可通过赋予$\n$坐标正权重的$\n$的伸缩轻松构造（给出一个由$n$个正整数组成的序列，我就能给出$\n$上向量场的一个可递幂零李代数）。有趣的是，所有向量场的可递幂零局部李代数都可作为这类幂零代数的子代数得到。从一个分次幂零李代数出发，人们可以归纳地构造其田中（Tanaka）延拓的分次部分，即“次数为0、1等的导子”。当然，相对于该伸缩的权重为$k$的向量场会自动定义权重为$k$的导子，因此这种情况下田中延拓永远不会是有限的。所有这类导子都由向量场给出，还是存在额外的“奇异”导子？我们回答了这个问题。除特殊情形外，次数为0的导子由次数为0的向量场给出，且田中延拓可恢复由该伸缩定义的整个多项式向量代数。然而，在某些特定的伸缩情形中，我们能找到“奇异”导子，并对其进行了详细描述。
英文摘要：
Transitive local Lie algebras of vector fields can be easily constructed from dilations of $\mathbb{R}^n$ associating with coordinates positive weights (give me a sequence of $n$ positive integers and I will give you a transitive nilpotent Lie algebra of vector fields on $\mathbb{R}^n$). It is interesting that all transitive nilpotent local Lie algebra of vector fields can be obtained as subalgebras of nilpotent algebras of this kind. Starting with a graded nilpotent Lie algebra one constructs graded parts of its Tanaka prolongations inductively as `derivations of degree 0, 1, etc. Of course, vector fields of weight $k$ with respect to the dilation define automatically derivations of weight $k$, so the Tanaka prolongation is in this case never finite. Are they all such derivations given by vector fields or there are additional `strange'ones? We answer this question. Except for special cases, derivations of degree 0 are given by vector fields of degree 0 and the Tanaka prolongation recovers the whole algebra of polynomial vectors defined by the dilation. However, in some particular cases of dilations we can find `strange' derivations which we describe in detail
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