论文标题
媒介场及其田中延长的传递nilpotent Lie代数
Transitive nilpotent Lie algebras of vector fields and their Tanaka prolongations
论文作者
论文摘要
可以轻松地从$ \ mathbb {r}^n $的扩张中构建的矢量场的局部谎言代数，与坐标的正权重相关联（给我$ n $阳性整数的顺序，我会给您带来的nilpotent nilpotent lie代数的vector of vector of vector of vector fields of $ \ mathbbbbbbbbbb {r r}^n $）。有趣的是，可以将矢量场的所有传递nilpotent局部谎言代数作为此类nilpotent代数的子代数获得。从分级的nilpotent Lie代数开始，一个构造其田中延长的部分构造部分，因为``0、1、1的推导''的衍生物，等等。当然，重量$ k $的矢量在扩张方面定义自动衍生$ k $，因此在这种情况下，Tanaka延长是无限的。它们都是向量场给出的这些推导，还是还有其他“奇怪”？我们回答这个问题。除特殊情况外，程度0的推导是由度量0的矢量场给出的，田中延长恢复了由扩张定义的多项式矢量的整个代数。但是，在某些特定的扩张情况下，我们可以找到我们详细描述的“奇怪”推导
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
