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My question is about the construction of Verma module of a lie algebra $L$, there is one step in the construction which I do not quite understand.

Let $L=N_-\oplus H\oplus N_+$ be the triangular decomposition of a lie algebra $L$ over the field $\mathbb{C}$. Let $U(L)$ be the universal enveloping algebra of $L$. Define $I_\lambda$ be the left idea of $U(L)$ generated by $N_+$ and $h-\lambda(h)1$:

$$I_\lambda=\{xn_++y(h-\lambda(h)1)\ |\ x,y\in U(L),n_+\in N_+,h\in H\}.$$

I want to show that $I(\lambda)\cap N_-=(0)$. This is easy at the first sight, but then I found it's subtle to use the PBW theorem directly: when assuming $x,y$ are standard PBW basis monomials, and substitute them into the expressions above, one does not get a standard linear combinations of standard monomials. Can anyone help me fix this problem?

zemora
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  • You may want to ask this on mathoverflow, as this is certainly graduate level mathematics. I've found I don't usually get good answers for Lie-theory questions on math.se – Mathmo123 Aug 17 '14 at 10:46
  • The answer is a little involved. But what you are trying to fight through is exact Proposition 10.5 (page 180) in Roger Carter's "Lie Algebras of Finite and Affine Type". If you don't have that text, I would highly recommend getting a copy. It's a truly excellent book. :) – Bill Cook Aug 20 '14 at 14:21

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