The Toda flow on Hessenberg elements of real, split simple Lie algebras

In this work, we consider the Toda flow associated with compact/Borel decompositions of real, split simple Lie algebras. Using the primitive invariant polynomials of Chevalley, we show how to construct integrals in involution which are invariants of the maximal compact subgroup, and moreover, we show that the number of such integrals is given by a formula involving only Lie-theoretic data. We then introduce the space of Hessenberg elements, characterize the generic Hessenberg coadjoint orbits, and show that the dimension of such orbits is precisely twice the number of nontrivial invariants which appeared earlier. For the class of classical, real split simple Lie algebras, we construct angle-type variables which in particular shows that the Toda flow is Liouville integrable on generic Hessenberg coadjoint orbits.

在这项工作中，我们考虑与实的、分裂单李代数的紧/博雷尔分解相关的托达流。利用谢瓦莱的本原不变多项式，我们展示了如何构造对合积分，这些积分是极大紧子群的不变量，而且我们表明此类积分的数量由一个仅涉及李理论数据的公式给出。然后我们引入黑森伯格元素空间，刻画一般的黑森伯格余伴随轨道，并表明此类轨道的维数恰好是前面出现的非平凡不变量数量的两倍。对于经典的、实分裂单李代数类，我们构造了角度型变量，这特别表明托达流在一般的黑森伯格余伴随轨道上是刘维尔可积的。