# Verma module

Let $\U0001d524$ be a semi-simple Lie algebra, $\U0001d525$ a Cartan subalgebra^{}, and
$\U0001d51f$ a Borel subalgebra. We work over a field $F$. Given a weight $\lambda \in {\U0001d525}^{*}$, let ${F}_{\lambda}$ be the 1-d dimensional $\U0001d51f$-module on which $\U0001d525$ acts by multiplication^{} by $\lambda $, and the positive root spaces act trivially. Now, the Verma module^{} ${M}_{\lambda}$ of the weight $\lambda $ is the $\U0001d524$-module induced from ${F}_{\lambda}$, i.e.

$${M}_{\lambda}=\mathcal{U}(\U0001d524){\otimes}_{\mathcal{U}(\U0001d51f)}{F}_{\lambda}.$$ |

Using the PoincarÃÂ©-Birkhoff-Witt theorem we see that as a vector space ${M}_{\lambda}$ is isomorphic^{} to $\mathcal{U}(\overline{\U0001d52b})$, where $\overline{\U0001d52b}$ is the sum of the negative weight spaces (so $\U0001d524=\U0001d51f\oplus \overline{\U0001d52b})$. In particular ${M}_{\lambda}$ is infinite dimensional.

We say a $\U0001d524$-module $V$ is a highest weight module if it has a weight $\mu \in {\U0001d525}^{*}$ and a non-zero vector $v\in {V}_{\mu}$ with $Xv=0$ for any $X$ in a positive root space and such that $V$ is generated as a $\U0001d524$-module by $v$. The Verma module ${M}_{\lambda}$ is a highest weight module and we fix a generator^{} $1\otimes 1$.

The most important property of Verma modules is that they are universal^{} amongst highest weight modules, in the following sense. If $V$ is a highest weight module generated by $v$ which has weight $\lambda $ then there is a unique surjective^{} homomorphism^{} ${M}_{\lambda}\to V$ which sends $1\otimes 1$ to $v$. That is, all highest weight modules with highest weight $\lambda $ are quotients of ${M}_{\lambda}$. Also, ${M}_{\lambda}$ has a unique maximal submodule, so there is a unique irreducible representation with highest weight $\lambda $. If $\lambda $ is dominant and integral then this module is finite dimensional.

Title | Verma module |
---|---|

Canonical name | VermaModule |

Date of creation | 2013-03-22 13:12:13 |

Last modified on | 2013-03-22 13:12:13 |

Owner | owenjonesuk (12024) |

Last modified by | owenjonesuk (12024) |

Numerical id | 8 |

Author | owenjonesuk (12024) |

Entry type | Definition |

Classification | msc 17B20 |

Defines | highest weight module |