论文标题
$ b $ - 谎言组的结构和减少泊松
$b$-Structures on Lie groups and Poisson reduction
论文作者
论文摘要
以固定时间为零的伽利略转型组和加利利转型的子组的动机，我们介绍了$ b $ - lie group的概念，作为一对$ $（g，h）$，其中$ g $是一个谎言组，$ h $是一个codimension-nie-lie子组。这样的概念使我们能够为时空转换提供一个理论框架，在该框架可以将初始时间视为边界。在这个理论框架中，我们开发了理论的基础，并研究了$ b $ -cotangengent bundle $^b {t}^\ ast g $上的相关规范$ b $ -symplectic结构以及其还原理论。也就是说，我们将最小的耦合过程扩展到$^bt^*g/h $，并证明可以用左翻译的cotangang cotangent升起的动作减少poisson，可以用$ \ mathfrak上的lie poisson结构来描述，$ \ mathfrak {H} $ b $ -symplectic结构上的$^b {t}^\ ast（g/h）$，其中$ g/h $被视为一维$ b $ -manifold，具有关键的hypersurface（从$ b $ - manifolds的意义上）。
Motivated by the group of Galilean transformations and the subgroup of Galilean transformations which fix time zero, we introduce the notion of a $b$-Lie group as a pair $(G,H)$ where $G$ is a Lie group and $H$ is a codimension-one Lie subgroup. Such a notion allows us to give a theoretical framework for transformations of space-time where the initial time can be seen as a boundary. In this theoretical framework, we develop the basics of the theory and study the associated canonical $b$-symplectic structure on the $b$-cotangent bundle $^b {T}^\ast G$ together with its reduction theory. Namely, we extend the minimal coupling procedure to $^bT^*G/H$ and prove that the Poisson reduction under the cotangent lifted action of $H$ by left translations can be described in terms of the Lie Poisson structure on $\mathfrak{h}^\ast$ (where $\mathfrak{h}$ is the Lie algebra of $H$) and the canonical $b$-symplectic structure on $^b {T}^\ast(G/H)$, where $G/H$ is viewed as a one-dimensional $b$-manifold having as critical hypersurface (in the sense of $b$-manifolds) the identity element.
