Abstract

By Cartan’s Theorem, every closed subgroup \(H\) of a real (or \(p\)-adic) Lie group \(G\) is a Lie subgroup. For Lie groups over a local field \({{\mathbb K}}\) of positive characteristic, the analogous conclusion is known to be wrong. We show more: There exists a \({{\mathbb K}}\)-analytic Lie group \(G\) and a non-discrete, compact subgroup \(H\) such that, for every \({{\mathbb K}}\)-analytic manifold \(M\), every \({{\mathbb K}}\)-analytic map \(f\colon M\to G\) with \(f(M)\subseteq H\) is locally constant. In particular, the set \(H\) does not admit a non-discrete \({{\mathbb K}}\)-analytic manifold structure which makes the inclusion of \(H\) into \(G\) a \({{\mathbb K}}\)-analytic map. We can achieve that, moreover, \(H\) does not admit a \({{\mathbb K}}\)-analytic Lie group structure compatible with the topological group structure induced by \(G\) on \(H\).

中文翻译：

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局部正特征场上李群中的非李子群

摘要

根据 Cartan 定理，实数（或\(p\) -adic）李群\(G\)的每个闭子群\(H\)都是李子群。对于具有正特征的局部域\({{\mathbb K}}\)上的李群，已知类似的结论是错误的。我们展示了更多： 存在一个\({{\mathbb K}}\) -分析李群\(G\)和一个非离散紧子群\(H\)使得对于每个\({{\ mathbb K}}\) -解析流形\(M\)，每个\({{\mathbb K}}\) -解析映射\(f\colon M\to G\)与\(f(M)\subseteq H\）是局部常数。特别是，集合\(H\)不允许非离散\({{\mathbb K}}\) -解析流形结构，这使得\(H\)包含在\(G\ )中({{\mathbb K}}\) -解析图。此外，我们可以实现，\(H\)不承认\({{\mathbb K}}\) -与由\(G\)在\(H\ ) .