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Given a simply connected locally compact group $G$, is it true that $G$ admits enough finite dimensional representations (over any field and not necessarily continuous) to separate points in $G$, what about over $\mathbb{C}$ and we require the representations to be continuous?

Again, this question is a follow-up of this one, and it seems better to ask it separately here.

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    $\begingroup$ Ben Wieland answered this in the negative in a comment to the original question, assuming continuity. This assumption is not needed really, by the work of Borel-Tits on "abstract homomorphisms". $\endgroup$
    – Uri Bader
    Commented Dec 12, 2021 at 8:33
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    $\begingroup$ The smallest counterexample is the universal covering of $\mathrm{SL}_2(\mathbf{R})$, which is even contractible (homeomorphic to $\mathbf{R}^3$). Possibly it is easier to prove non-linearity (without continuity) for the 2-fold covering of $\mathrm{SL}_n(\mathbf{R})$ for larger $n$. These contain f.g. non-residually-finite subgroups, but I'd like an elementary argument for this. $\endgroup$
    – YCor
    Commented Dec 12, 2021 at 11:04
  • $\begingroup$ Please see the article ams.org/journals/tran/1980-259-02/S0002-9947-1980-0567087-9 $\endgroup$
    – Onur Oktay
    Commented Jan 16, 2022 at 17:18

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The connected Lie groups whose points are separated by the finite-dimensional complex representations are exactly the linear Lie groups, for instance by Th. 5.3 in Beltiţă and Neeb - Finite-dimensional Lie subalgebras of algebras with continuous inversion.

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