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is there a natural way to associate a Lie algebra (over real numbers or other field) to a profinite group in a functorial way ? I.e. I'm looking for a functor $L: \mathbf{ProfinGroups}\rightarrow \mathbf{LieAlg}$

If such functor exists, could we have a fully faithful functor if we restrict $L$ to some full subcategory of $\mathbf{ProfinGroups}$ ?

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  • $\begingroup$ Can you do that for a finite group? $\endgroup$
    – Denis Nardin
    Commented Jan 3, 2017 at 21:21
  • $\begingroup$ @DenisNardin that is part of the question! $\endgroup$
    – Ofra
    Commented Jan 3, 2017 at 21:25
  • $\begingroup$ Yes there exists a functor. You need some further conditions to make the question nontrivial. $\endgroup$
    – YCor
    Commented Jan 3, 2017 at 21:55
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    $\begingroup$ @YCor I'm looking for those conditions! I'm in learning stage. Any help will be appreciated $\endgroup$
    – Ofra
    Commented Jan 3, 2017 at 22:59
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    $\begingroup$ Read Serre's book "Lie groups and Lie algebras" to see what is done for $p$-adic Lie groups (which are a special class of profinite groups near the identity); this will also make clearer the variant one should consider in place of full faithfulness given that connectedness is not available. $\endgroup$
    – nfdc23
    Commented Jan 4, 2017 at 8:24

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