Definition of Verma modules

I have a question regarding different (but equivalent!?) definitions of Verma modules of semisimple Lie algebras:

Let F be a field and denote the following:

• $ \mathfrak{g}$ , a semisimple Lie algebra over F, with universal enveloping algebra $\mathcal{U}(\mathfrak{g})$.
• $\mathfrak{b}$, a Borel subalgebra of $\mathfrak{g}$, with universal enveloping algebra $\mathcal{U}(\mathfrak{b})$.
• $\mathfrak{h}$, a Cartan subalgebra of $\mathfrak{g}$.
• $\lambda \in \mathfrak{h}^*$, a fixed weight.

1.Definition (von Wikipedia)

Let be $F_\lambda$, the one-dimensional F-vector space (i.e. whose underlying set is F itself) together with a $\mathfrak{b}$-module structure such that $\mathfrak{h}$ acts as multiplication by $\lambda$ and the positive root spaces act trivially. As $F_\lambda $is a left $\mathfrak{b}$-module, it is consequently a left $\mathcal{U}(\mathfrak{b})$-module. Using the Poincaré-Birkhoff-Witt theorem, there is a natural right $\mathcal{U}(\mathfrak{b})$-module structure on $\mathcal{U}(\mathfrak{g})$ by right multiplication of a subalgebra. $\mathcal{U}(\mathfrak{g})$ is naturally a left $\mathfrak{g}$-module, and together with this structure, it is a $(\mathfrak{g}, \mathcal{U}(\mathfrak{b}))$-bimodule.

Now we can define the Verma module (with respect to $\lambda$) as

$$ M_\lambda = \mathcal{U}(\mathfrak{g}) \otimes_{\mathcal{U}(\mathfrak{b})} F_\lambda $$

which is naturally a left $\mathfrak{g}$-module (i.e. an infinite-dimensional representation of $\mathfrak{g})$.

2.Definition

Let be $R^+$ the set of positive roots and $g_\alpha$ the root space to $\alpha \in R^+$. Let be $I_\lambda$ the left ideal of $U(g)$ which is generated from all $X \in g_\alpha$ for some $\alpha \in R^+$ and from all $H-\lambda(H)$ with $H\in \mathfrak{h}$.

We define now the Verma module

$$V_\lambda=\mathcal{U}(\mathfrak{g})/I_\lambda$$

My question is now, why these definition are equivalent? I tried quite a while, but I'm not very familiar with this and don't really know how to proceed. To you have some hints?

Thanks in advance!

Construct the left $\mathfrak{g}$-module homomorphism $$ \begin{align} \mathcal{U}(\mathfrak{g}) \quad &\to \quad \mathcal{U}(\mathfrak{g}) \otimes_{\mathcal{U}(\mathfrak{b})} F_{\lambda} \\ U \quad &\mapsto \quad U \otimes 1_{\lambda}. \end{align} $$ Verify that it is surjective and that the kernel is $I_{\lambda}$.

Hint: Since $F_{\lambda}$ has a single generator, it is isomorphic as a $\mathcal{U}(\mathfrak b)$ module to $\mathcal{U}(\mathfrak b)/I_{\lambda}$ for some ideal $I_{\lambda}$.