# nLab Verma module

Contents

### Context

#### Lie theory

Background

Smooth structure

Higher groupoids

Lie theory

∞-Lie groupoids

∞-Lie algebroids

Formal Lie groupoids

Cohomology

Homotopy

Related topics

Examples

$\infty$-Lie groupoids

$\infty$-Lie groups

$\infty$-Lie algebroids

$\infty$-Lie algebras

# Contents

## Idea

Given a semisimple Lie algebra $\mathfrak{g}$ over a field $F$ with Borel subalgebra? $\mathfrak{b}\subset\mathfrak{g}$ a Verma module is the induced module

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

where $F_\lambda$ is one dimensional representation of $U(\mathfrak{b})$ corresponding to the character $\lambda$ of $\mathfrak{b}$.

This construction is used also for many other Lie algebras with triangular decomposition, for quantized enveloping algebras and many other generalizations.

Last revised on August 17, 2022 at 09:31:30. See the history of this page for a list of all contributions to it.