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I'm curious about the representation theory of solvable Lie algebras. Consider the following quote from Fulton & Harris:

[To] study the representation theory of an arbitrary Lie algebra, we have to understand individually the representation theories of solvable and semisimple Lie algebras. Of these, the former is relatively easy, at least as regards irreducible representations. The basic fact about them—that any irreducible representation of a solvable Lie algebra is one dimensional—will be proved later in this lecture.

The phrasing of the bold part suggests that, while the irreps are very easy to describe (they are all one dimensional), the situation for general reps is still "relatively easy," perhaps only slightly more complicated. On the other hand, we can't expect to be able to just build up all reps from irreps by taking direct sums, as is the case for semisimple Lie algebras.

So, to what extent is the bold part true? In particular, what are the main results (if any) summarizing the representation theory of solvable Lie algebras?

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  • $\begingroup$ I think the way to read that sentence is that specifically the irreps are relatively easy to describe. It perhaps gives the impression that the whole theory is straightforward but technically it doesn't say that, it is only claiming that the irreps are. $\endgroup$
    – Callum
    May 16 at 21:11

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Finite-dimensional irreducible representations of finite-dimensional complex solvable Lie algebras are $1$-dimensional by Lie's theorem. This is false for arbitrary irreducible representations, see here:

Are all irreducible representations of solvable Lie algebras 1-dimensional?

Furthermore, this does not imply that we can "classify" representations of solvable Lie algebras. This is impossible. It is much easier to study representations of semisimple Lie algebras.

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  • $\begingroup$ So in other words, rep theory of solvable algebras is only easy as far is irreps are concerned. I guess the claim in F&H is a little misleading then. $\endgroup$
    – WillG
    May 15 at 17:18
  • $\begingroup$ @WillG Yes, Fulton Harris should have said a bit more here. Where do they state this in the book? $\endgroup$ May 15 at 18:22
  • $\begingroup$ In my version (first edition), it's on page 124, in section 9.1. $\endgroup$
    – WillG
    May 16 at 5:00

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