主丛自同构群上的几何动力学：测地流、对偶对和色态群
Geometric dynamics on the automorphism group of principal bundles: geodesic flows, dual pairs and chromomorphism groups
摘要：
我们在主丛P的自同构的李群Aut（P）上构造Euler-Poincaré方程，相应的流称为EPAut流.本文主要研究与Kaluza-Klein型拉格朗日量有关的测地流。在一个平凡丛P的特殊情况下，我们确定测地线的某些无限维半直积李群，自然出现的建设。这种方法自然导致一个对偶对结构包含δ-like动量映射的解决方案，推广以前的结果测地流上的仿射群（EPDiff）。在第二部分中，我们考虑了主丛的保体积自同构李群上的不可压缩流。在这种情况下，对偶对的构造需要色态群的定义，即适当的李群扩张推广了量子态群。
英文摘要：
We formulate Euler-Poincaré equations on the Lie group Aut(P) of automorphisms of a principal bundle P. The corresponding flows are referred to as EPAut flows. We mainly focus on geodesic flows associated to Lagrangians of Kaluza-Klein type. In the special case of a trivial bundle P, we identify geodesics on certain infinite-dimensional semidirect-product Lie groups that emerge naturally from the construction. This approach leads naturally to a dual pair structure containing δ-like momentum map solutions that extend previous results on geodesic flows on the diffeomorphism group (EPDiff). In the second part, we consider incompressible flows on the Lie group of volume-preserving automorphisms of a principal bundle. In this context, the dual pair construction requires the definition of chromomorphism groups, i.e. suitable Lie group extensions generalizing the quantomorphism group.
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