 Verma module

For a Borel subalgebra $\mathfrak{b}=\mathfrak{t}\oplus {\mathfrak{n}}^{+}$ and a weight $\lambda$, the Verma module iswhere ${\mathfrak{n}}^{+}$ kills ${\mathbb{C}}_{\lambda }$ and $h\in \mathfrak{t}$ acts by $\lambda (h)$.

A Verma module is a type of representation of a highest weight module in the context of the representation theory of Lie algebras, particularly those that are semisimple. Verma modules play an important role in the study of the structure and representation theory of Lie algebras and quantum groups.