# Questions tagged [verma-modules]

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13 questions
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### Simple formula for the dimension of weight spaces of Verma module?

Let $\mathfrak{g}$ be a simple Lie algebra (e.g. $\mathfrak{sl}_n$), and let $M_\lambda$ be the Verma module with highest weight $\lambda$. Is there a simple formula for $\dim (M_\lambda)_\mu$, where ...
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### Extension group between Verma modules for $\mathfrak{sl}_2$

Let $\mathcal{g}=\mathfrak{sl}_2(\mathbb C)$ be the simple lie algebra of $\text{SL}_2$, for any $\lambda \in \mathbb C$ one can consider the corresponding Verma module $M_{\lambda}$, which is ...
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### Verma module analogue for non-semisimple Lie algebras

If $\mathfrak{g}$ is semisimple and complex, then we can identify a Cartan subalgebra $\mathfrak{h}$ and with choice of positive roots, a Borel subalgebra containing this Cartan. We can then consider ...
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### Composition series for Verma modules.

Let $L$ a Lie Algebra. I need prove that that every Verma module $\Delta(\lambda)$ admits a composition series, i.e a series of submodules with simple factors. I found a proof that is quite short in ...
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### How to find the multiplicity of weight in a Verma module?

In particular, let $\mathfrak g$ be the semisimple Lie algebra of type $A_{2}$ et let $\alpha,\beta$ be its simple roots. How can the multiplicity of weight $-2\alpha -3\beta$ be calculated in the ...
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 ...