nLab
Verma module

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.

Related concepts

• BGG resolution