Notes on regularity properties of infinite-dimensional Lie groups
Notes on regularity properties of infinite-dimensional Lie groups
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关于无限维李群正则性质的注解
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发表时间：
2012
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通讯作者：
Helge Glockner
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作者：
Helge Glockner
令 G 为在局部凸空间上建模的李群，李代数为 g，k 为非负整数或无穷大。如果 g 中的每条 C^k 曲线 c 都承认 G 中的左演化 Evol(c)，则我们称 G 是 C^k 半正则。此外，如果 c 到 evol(c):=Evol(c)(1) 的映射是平滑的，则 G 称为 C^k 正则。对于 G 是 C^k 半正则李群，m 是可微分阶，我们证明 evol 是 C^m 当且仅当 Evol 是 C^m 时。如果 evol 在 0 处连续，则 evol 是连续的。如果 G 是 C^0 半正则李群，则 evol 的连续性意味着其平滑性（因此 G 将是 C^0 正则），如果从 G 到 C^0 正则李群的平滑同态将 G 上的点分开，并且 g 是（例如）顺序完备的。提供了正则性性质的进一步标准，并用于证明几类重要的李群的正则性。值得注意的是，我们发现仿紧有限维光滑流形 M（不需要是 sigma 紧）的光滑微分同胚的李群 Diff(M) 是 C^1 正则。我们还提供了能够证明紧实解析流形的解析微分同胚李群是 C^1 正则的工具。
Let G be a Lie group modelled on a locally convex space, with Lie algebra g, and k be a non-negative integer or infinity. We say that G is C^k-semiregular if each C^k-curve c in g admits a left evolution Evol(c) in G. If, moreover, the map taking c to evol(c):=Evol(c)(1) is smooth, then G is called C^k-regular. For G a C^k-semiregular Lie group and m an order of differentiability, we show that evol is C^m if and only if Evol is C^m. If evol is continuous at 0, then evol is continuous. If G is a C^0-semiregular Lie group, then continuity of evol implies its smoothness (so that G will be C^0-regular), if smooth homomorphisms from G to C^0-regular Lie groups separate points on G and g is (e.g.) sequentially complete. Further criteria for regularity properties are provided, and used to prove regularity for several important classes of Lie groups. Notably, we find that the Lie group Diff(M) of smooth diffeomorphisms of a paracompact finite-dimensional smooth manifold M (which need not be sigma-compact) is C^1-regular. We also provide tools which enable to show that the Lie group of analytic diffeomorphisms of a compact real analytic manifold is C^1-regular.