论文标题
泊松群岛上哈密顿动力学的单二型性和不变体积形式
Unimodularity and invariant volume forms for Hamiltonian dynamics on Poisson-Lie groups
论文作者
论文摘要
在本文中，我们讨论了哈密顿系统在泊松群体上的不变体积形式的存在与泊松结构的单对象之间的关系。特别是，我们证明了一个谎言群体上的hamiltonian矢量场，并具有单模型的泊松结构，保留了该组上任何剩余体积的倍数。相反，我们还证明，如果存在哈密顿量的功能，以使谎言组的身份元素是非排定奇异性，而相关的哈密顿载体场则保留了体积形式，那么泊松结构必定是单模型的。此外，我们以不同的有趣示例来说明我们的理论，无论是在半神经和单模型的泊松群体上。
In this paper, we discuss several relations between the existence of invariant volume forms for Hamiltonian systems on Poisson-Lie groups and the unimodularity of the Poisson-Lie structure. In particular, we prove that Hamiltonian vector fields on a Lie group endowed with a unimodular Poisson-Lie structure preserve a multiple of any left-invariant volume on the group. Conversely, we also prove that if there exists a Hamiltonian function such that the identity element of the Lie group is a nondegenerate singularity and the associated Hamiltonian vector field preserves a volume form, then the Poisson-Lie structure is necessarily unimodular. Furthermore, we illustrate our theory with different interesting examples, both on semisimple and unimodular Poisson-Lie groups.
