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I have heard that the branching rules are well-known for the simple Lie algebra $\mathfrak{s}\mathfrak{l}_n$ in $\mathfrak{s}\mathfrak{l}_{n+1}$ over fields of characteristic zero. Where can I find a thorough treatment of this result?

As a secondary question, are there any papers that deal with "branching rules" when the subalgebra is not semisimple? Obviously we no longer have Weyl's theorem so the notion of branching rules may not quite make sense, but I am wondering what sort of results are there when we restrict irreducible representations to non-semisimple subalgebras. Again, this is over fields of characteristic zero.

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  • $\begingroup$ To your secondary question, if the subalgebra is not semisimple, its representation theory might be very mysterious and have nothing to do with weights, roots, ... to begin with! $\endgroup$
    – Torsten Schoeneberg
    Commented Jul 25, 2019 at 21:05

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