Consider all the finite-dimensional irreducible representations of a group.

  1. For each finite-dimensional irreducible representation of a group, is there one and only one corresponding representation of the Lie algebra?

  2. For a Lie group, are there always finite-dimensional irreducible representations of $n \times n$ matrices for all values of $n$?

  3. For a Lie group, can there only be one finite-dimensional irreducible representation of $n \times n$ matrices for each value of n?


closed as off-topic by Daniel Robert-Nicoud, G-man, Michael Albanese, Cameron Williams, Mark Viola Nov 20 '15 at 0:38

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  • $\begingroup$ For 1, for non simply connected Lie groups, representations of the Lie algebra don't generally lift to the group. So I would say no. Look for instance at $G=S^1$, $\frak{ g}=\mathbb{R}$ acting on $\mathbb{R}$. $\endgroup$ – Tim kinsella Nov 18 '15 at 20:24
  1. Every representation of a Lie group $G$ naturally induces a corresponding representation of the corresponding Lie algebra $\mathfrak{g}$, by differentiation. I'm not sure how to interpret "only one" in this question.

  2. No. For example, $SO(3)$ has irreducible representations only in odd dimensions $1, 3, 5, \dots$.

  3. No. For example, $SU(2) \times SU(2)$ has two nonisomorphic irreducible $2$-dimensional representations, one for each of its factors.


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