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I would like to ask about the cardinality of the sets of irreducible, inequivalent representations of Lie groups or Lie algebras. I will use the term irreps (of a group/algebra) to refer to finite-dimensional, inequivalent and irreducible representations.

I know that, for example, $SO(3)$ has irreducible representations only in odd dimensions (hence inequivalent). I imagine therefore the set of irreps is countable (correct?).

Then, there are irreps of $SU(2)$ (equivalently, $\mathfrak{su}(2)$, or its complexification $\mathfrak{su}(2)_{\mathbb{C}} = \mathfrak{sl}(2,\mathbb{C}$)), which are indexed, e.x. by physicists, by $s\in \frac{1}{2}\mathbb{Z_{+}}$, acting on a vector space of polynomials in two complex variables. Therefore, there are infinitely countably many irreps.

  1. Are there some easy to list general results about such classification, whether we concern ourselves with:

a) finitely dimensional representations, or

b) infinitely-dimensional representations?

  1. In the general case, is the set of irreps of compact/noncompact Lie group or a Lie algebra infinite, and what is its cardinality?

And in the case of negative answer to 2.:

  1. Can there be finitely many irreps for either a compact or a non-compact Lie group?

I have not found clear and concise statements in the literature and would be grateful for suggestions.

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  • $\begingroup$ You mean representations on complex vector spaces, I assume? (Or at least some characteristic 0 field, otherwise things get fishy.) $\endgroup$ Jul 12, 2020 at 16:38
  • $\begingroup$ Re question 3, have a look at finite groups, or $\mathbb R$, or $\mathbb R^*$, or $\mathbb C^*$. $\endgroup$ Jul 12, 2020 at 16:40
  • $\begingroup$ @Torsten Schoeneberg, yes, I meant characteristic 0. Oh, that was so straightforward actually, thanks $\endgroup$
    – K.T.
    Jul 12, 2020 at 17:44
  • $\begingroup$ To qualify my earlier comment though, irreps of the abelian Lie groups, although all of dimension 1, can come in families of continuum cardinality. $\endgroup$ Jul 12, 2020 at 22:40

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All irreducible representations of a compact Lie group are finite-dimensional, and a compact Lie group has countably-many irreducible representations. This follows for example from my answer here. For a compact Lie group you are not going to have only finitely-many irreducible representations unless $L^2(G)$ is finite-dimensional. This will not happen unless $G$ is finite.

For non-compact Lie groups there will be in general infinite-dimensional irreducible representations, and continuum-many irreducible representations. The more interesting thing to do is sort them into various families each depending, say, on a single real parameter. This is due in greatest part to Harish-Chandra and the philosophy he developed, and is many decades of work starting in the late 1940s, essentially, with Bargmann's classification of unitary irreducible representations of $SL_2(\mathbb{R})$ (and it's rather easy to see using Weyl's unitary trick that there are no finite-dimensional unitary representations at all in this case).

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