平凡混合阶常微分方程组的对称性
Symmetries of trivial systems of ODEs of mixed order
摘要：
我们计算两个方程组y^（k）=z^（l）=0的对称代数，其中2<=k<l。看来有许多方法可以将这样的常微分方程系统转化为外微分系统。它们导致不同系列的有限维对称代数。例如，对于（k，l）=（2，3），我们得到两个相同维数的非同构对称代数。我们探讨了这些对称代数与G-结构的Sternberg延拓和分次幂零李代数的Tanaka延拓之间的关系。令人惊讶的是，情况（k，l）=（2，3）提供了gl（5，R）中线性子代数g的一个例子，使得g和g^t的斯滕伯格交换都具有相同的维数，但不同构。我们还讨论了非线性的情况和与光滑流形上旗结构的联系。
英文摘要：
We compute symmetry algebras of a system of two equations y^(k)=z^(l)=0, where 2<=k<l. It appears that there are many ways to convert such system of ODEs to an exterior differential system. They lead to different series of finite-dimensional symmetry algebras. For example, for (k,l)=(2,3) we get two non-isomorphic symmetry algebras of the same dimension. We explore how these symmetry algebras are related to both Sternberg prolongation of G-structures and Tanaka prolongation of graded nilpotent Lie algebras. Surprisingly, the case (k,l)=(2,3) provides an example of a linear subalgebra g in gl(5,R) such that the Sternberg prolongations of g and g^t are both of the same dimension, but are non-isomorphic. We also discuss the non-linear case and the link with flag structures on smooth manifolds.
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