Questions tagged [verma-modules]

Question 1

I'm trying to prove the next result from Kac's book, Infinite dimensional Lie algebras. Let $V$ be a $\mathfrak{g}(A)$-module with highest weight $\Lambda$. Then $$ \text{ch}(V)=\sum_{\lambda\in B(\...

Question 2

I’m studying Verma modules of $\mathfrak{sl}_{2} = \mathfrak{sl}_{2} (\mathbb{C})$. Let’s introduce standard notation. Elements $h,e,f\in \mathfrak{sl}_{2} $ form the basis of $\mathfrak{sl}_{2}$, ...

Question 3

Let $T \subset B \subset G$ be a reductive group, a Borel subgroup and a maximal torus over a field of characteristic $0$, with respective Lie algebras $\mathfrak{b} \subset \mathfrak{g}$. For a ...

Question 4

Let $L_m$ be the $m+1$ dimensional simple module over $\mathfrak{sl}_2(\Bbb{C})$, and $M_\lambda,\; \lambda\in \Bbb C$, the Verma module with highest weight $\lambda$ over $\frak{sl}_2(\Bbb C)$. Show ...

Question 5

I am studying Verma modules (reading Dixmier´s Enveloping Algebras) and have a question regarding the definition as a quotieng of the enveloping algebra. Let $g$ be a Lie algebra and $h$ its Cartan ...

Question 6

It is easy to show that every irreducible finite-dimensional real $\mathfrak{sl}_2(\mathbb C)$-representation is a weight module. The operator commonly denoted as $h$ in $\mathfrak{sl}_2$ has an ...

Question 7

What is the intuition behind verma modules? Their construction is quite technical to me and I know they help classify irreducible representations and are very useful, but I cannot see how. In class, ...

Question 8

For a semisimple Lie algebra $\mathfrak{g}$ and a Cartan subalgebra $\mathfrak{h}$, if $\mu \in \mathfrak{h}$ and $W_{\mu}$ is the associated Verma module, I want to see that the multiplicity of $\...

Question 9

I want to prove the following result: Theorem: Let $\mathfrak{g}$ be a semi-simple Lie algebra, $\mathfrak{h}$ be a Cartan subalgebra, $\mu \in \mathfrak{h}$, and $W_{\mu} = \mathfrak{U}_{\mathfrak{g}...

Question 10

I'm starting to look into Lie Superalgebras and understand a bit more about simple f.d. modules. Right now I'm studying fairly simple examples to try and get a feel for it. I'm trying to find the ...

Question 11

The character formula of a Verma module over a Kac-Moody algebra is given by $$\textrm{ch}\ M(\Lambda)=\frac{e(\Lambda)}{\prod_{\alpha\in\Phi+}(1-e(-\alpha))^{\textrm{mult}(\alpha)}}$$ Here $\Phi_+$ ...

Question 12

I have come across the terms "basic representation" of a semisimple Lie algebra, but I am finding it hard to find a clear definition of this representation. Can someone provide me a ...

Question 13

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 ...

Question 14

Let $\mathfrak{g}(A)$ be a symmetrizable Kac-Moody Lie algebra over $\mathbb{C}$ and ($\mathfrak{h}$, $\Pi, \Pi^\vee)$ be a realization of the GCM $A$. Assume that $$\mathfrak{g}(A)=\mathfrak{h} \...

Question 15

Let $g$ be a semi simple Lie algebra, $\rho$ be the half sum of all positive roots. I want to show that The Verma module $M(-\rho)$ is irreducible I have no idea how to approache this. Any hint or ...

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Questions tagged [verma-modules]

Character of a kac-moody module as the sum of characters of Verma modules

A particular quotient in the study of tensor products of $\mathfrak{sl}_2$-modules

Verma modules as representations of the Borel

Filtration of Verma modules tensored with a simple module

Question about definition of Verma modules

Irreducible finite-dimensional real $\mathfrak{sl}_2(\mathbb R)$-representations

Intuition behind verma modules

Counting the multiplicity of weight of a Verma module without using the Kostant partition function or the Weyl character formula.

Constructing a surjective intertwining map onto Verma module

Simple quotient of Verma Module $ M_\mathfrak{b}(\lambda) $

Character formula of Verma module over a Kac-Moody algebra

Basic representation of a simple Lie algebra and its highest weight

Equivalence of verma modules

Embedding of Verma modules in Kac-Moody Lie algebras

How to show a Verma module is irreducible

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