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.

  • 1
    $\begingroup$ 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
    Dec 12, 2021 at 8:33
  • 1
    $\begingroup$ 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
    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
    Jan 16, 2022 at 17:18

1 Answer 1


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.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.