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Non-Lie Subgroups in Lie groups over Local Fields of Positive Characteristic

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 \(H\).

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Acknowledgments

The author thanks C. R. E. Raja (Indian Statistical Institute, Bangalore) for questions and discussions which inspired the work. The referee’s comments helped to improve the presentation.

Funding

Supported by Deutsche Forschungsgemeinschaft, project GL 357/10-1.

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Correspondence to Helge Glöckner.

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Glöckner, H. Non-Lie Subgroups in Lie groups over Local Fields of Positive Characteristic. P-Adic Num Ultrametr Anal Appl 14, 138–144 (2022). https://doi.org/10.1134/S2070046622020042

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  • DOI: https://doi.org/10.1134/S2070046622020042

Keywords

  • Lie group
  • local field
  • positive characteristic
  • Cartan’s theorem
  • subgroup
  • Lie subgroup
  • submanifold
  • initial Lie subgroup
  • compatible Lie group structure