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Theorem 19.1b of Humphreys' Introduction to Lie Algebras and Representation Theory states that if a complex Lie algebra has a finite-dimensional non-trivial (Update: faithful) irreducible representation, then it is reductive. Unfortunately, a proof is not readable. Could you explain the idea?

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Actually, Theorem $19.1 (b)$ says something different, namely the following: if a Lie algebra $L$ of linear transformations in $\mathfrak{gl}(V)$ acts irreducibly on $V$ then $L$ is reductive with $0$ or $1$-dimensional center. The proof relies on Lie's Theorem showing that the solvable radical $rad(L)$ of $L$ consists only of the center. But a Lie algebra $L$ which satisfies $Rad(L)=Z(L)$, or $L=[L,L]\oplus Z(L)$ is reductive. This has been explained on MSE here.

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  • $\begingroup$ It is probably also worth pointing out that version of the theorem the OP stated is quite far from being true. $\endgroup$ – Tobias Kildetoft Feb 18 '17 at 13:19
  • $\begingroup$ So the difference is that only the image is reductive, that is, does the word faithful makes the statement true? $\endgroup$ – evgeny Feb 18 '17 at 13:35

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