I have studied Lie algebras at the level of Humphrey's Introduction to Lie Algebras and Representation Theory. This only really includes representation theory of semisimple lie algebras. In the context of representations and in particular crystals I was wondering how do the representations of $\mathfrak{gl_n}$ relate to those of $\mathfrak{sl_n}$. Is there any suggested books or references I could look at?
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4$\begingroup$ Assuming we're in characteristic $0$ (or not dividing $n$), we have $\mathfrak{gl}_n = \mathfrak{sl}_n \oplus \mathfrak{gl}_1$, so an irreducible representation of $\mathfrak{gl}_n$ is just a pair of an irreducible representation $V$ of $\mathfrak{sl}_n$ and a weight by which $\mathfrak{gl}_1$ acts on $V$. The same holds for any reductive Lie algebra (over a field of characteristic $0$), which is the sum of its derived Lie algebra and its maximal central, toral subalgebra. $\endgroup$– LSpiceCommented Jan 23 at 17:02
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$\begingroup$ @LSpice For finite groups there is a (non constructive) proof via characters that representations of a direct sum are exactly tensor product of representations. Is there something similar for reducible lie algebras? That would exactly reduce the problem to that of semisimple lie algebras. Sorry if this is well known my algebra background is not very broad I am an undergrad. Thanks $\endgroup$– SmithCommented Jan 24 at 18:28
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1$\begingroup$ Re, it's a safe bet that most things that can be proved about representations of finite groups can be proved in the setting of reductive groups or Lie algebras in characteristic $0$, too. I don't know a specific reference for Lie algebras, though. Nonetheless, in this case it's easy: since the action of the centre intertwines that of the derived subalgebra, it must preserve invariant subspaces, so an irreducible representation of the full algebra remains irreducible upon restriction to the derived subalgebra. $\endgroup$– LSpiceCommented Jan 24 at 20:58
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