Are the branching rules the same for a lie algebra and it's corresponding lie group?

Question

I am interested to know if lie algebras of $su(n)$, $so(n)$ and $sp(n)$ have the same branching rules as groups $SU(n)$, $SO(n)$ and $Sp(n)$. I am studying Finite-Dimensional Lie Algebras and Their Representations for Unified Model Building, by Naoki Yamatsu, and I want to know if I can have the same rules for the groups mentioned above.

Answer 1

I love this article, but it uses a some-what sloppy convention that is very common in the physics literature: whenever they talk about a group they actually mean the corresponding Lie algebra.

So that answers your question in somewhat vacuous way.

More general (but only in the finite dimensional case!): the representations of the Lie algebra are all the ones that exist. Now two groups (e.g. $SO(3)$ and $SU(2)$) can have the same Lie algebra but not the same representations. Concretely in the example: all irreps of $su(n)$ are irreps of $SU(n)$ but only the odd-dimensional ones are also reps of $SO(3)$. In all cases however the group reps are a subset of the Lie algebra reps: given a lie algebra rep you have to check if it 'lifts' to the group, the other direction (from group rep to Lie algebra rep) always goes through.

The situation for branching between groups is similar: we get the branching rule on the Lie algebra level and then in concrete cases we have to see if they lift to the group level. However: if we know that a given lie algebra rep is a group rep for the big group then it is also a rep for the subgroup simply by restricting the action to the subgroup.

Thus we get a Lie algebra rep for the lie algebra of the subgroup of which the branching rule dictates how it decomposes as a sum of irreps and then it follows from the fact that we got this non-irreducible rep for the (small) Lie algebra by differentiating an actually existing group representation (of the small group) that each of the irreducible reps into which it decomposes also come from actually existing group representations, i.e. lift to the small group.