Real special orthogonal group of dimension 3

Irreps of SO(3)

The Lie algebra has dimensional irreps for , while the group only has irreps for . lie

Proof

Let be the basis defined in Lie algebra, and be the quadratic Casimir element. Now consider a representation on a finite-dimensional vector space , which we will invoke implicitly. Let such that

Then

are Ladder operators of . It follows that transforms in the same irrep as . Since is finite dimensional this must terminate at both ends, hence there exist such that

In addition since

it follows

and thus

hence

and since we have and . Now since , we have dimensional irreps of labelled by . Now assume has a corresponding group representation . Then

which is a contradiction unless and thus .¹

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Footnotes

1. 2023. Groups and representations, §6.8, pp. 93–96 ↩