Let $R^+$ be the set of positive roots and $L_\alpha$ the root space to $\alpha \in R^+$. $\mathfrak{h}$ is a Cartan subalgebra of L.
Let $I_\lambda$ be the left ideal of $U(L)$ which is generated by elements of the form $X \in L_\alpha$ for some $\alpha \in R^+$ and of the form $H-\lambda(H)$ with $H\in \mathfrak{h}$.
We define the Verma module
$$V(\lambda)=\mathcal{U}(\mathfrak{g})/I_\lambda$$
Let $F_\lambda$ be the one-dimensional F-vector space for $L_+=\oplus_{\alpha>0}L_{\alpha} \oplus H$ with action $h1=\lambda (h)1$ and $L_{\alpha}1=0, \forall \lambda >0$
I want to prove that $V(\lambda ) \cong U(L)\otimes_{U(L_+)}F_{\lambda}.$
So I define a left L-module homomorphism:
$$\phi : U(L)\to U(L)\otimes_{U(L_+)}F_{\lambda}, U \mapsto U\otimes 1_{\lambda}$$.
It is obvious that $I_{\lambda} \subset Ker\phi$, but I have trouble proving $I_{\lambda} = Ker\phi$.
Can you provide some hints? Thank you!
