真群胚与动量映射：线性化、仿射与凸性
Proper Groupoids and Momentum Maps: Linearization, Affinity and Convexity
摘要：
我们证明了真李群胚是局部可线性化的。因此，一个真李群胚的轨道空间是一个光滑的轨道空间（一个局部看起来像向量空间被线性紧李群作用所商的T2空间）。在真（拟）辛群胚的情形下，轨道空间具有自然的积分仿射结构，使其成为具有局部凸多面体边界的仿射轨道折叠，并且每个边界点附近的局部结构同构于紧李群的Weyl腔.然后，我们将这些结果的研究的哈密顿作用的适当的（准）辛群胚的动量映射，并表明，这些动量映射保持自然的横向仿射结构的局部凸性。许多文献中的凸性定理可以从这最后一个陈述和一些关于仿射映射的基本结果中恢复。
英文摘要：
We show that proper Lie groupoids are locally linearizable. As a consequence, the orbit space of a proper Lie groupoid is a smooth orbispace (a Hausdorff space which locally looks like the quotient of a vector space by a linear compact Lie group action). In the case of proper (quasi-)symplectic groupoids, the orbit space admits a natural integral affine structure, which makes it into an affine orbifold with locally convex polyhedral boundary, and the local structure near each boundary point is isomorphic to that of a Weyl chamber of a compact Lie group. We then apply these results to the study of momentum maps of Hamiltonian actions of proper (quasi-)symplectic groupoids, and show that these momentum maps preserve natural transverse affine structures with local convexity properties. Many convexity theorems in the literature can be recovered from this last statement and some elementary results about affine maps.
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