Triangular decomposition of a Lie algebra. Let \gLie be a Lie algebra over \bbk. A triangular decomposition of \gLie is a decomposition

\begin{aligned} \gLie = \gLie^0 \oplus \gLie^+ \oplus \gLie^- \end{aligned}

of \gLie into a vector space direct sum of three vector subspaces: Cartan part (or Cartan subalgebra) \gLie^0\subseteq \gLie, the upper part \gLie^+ \subseteq \gLie, and the lower part \gLie^- \subseteq \gLie.

(Note that in this definition we do not yet require \gLie^0, \gLie^+, \gLie^- \subseteq \gLie to be Lie subalgebras, with the Cartan subalgebra being abelian and the upper and lower parts being nilpotent. In intended use cases, we typically have these properties, however.)

Verma module. Let \gLie be a Lie algebra over \bbk, with a triangular decomposition \gLie = \gLie^0 \oplus \gLie^+ \oplus \gLie^-. Let \eta \colon \gLie^0 \to \bbk be a linear functional on the Cartan part. Then the Verma module associated to the "highest weight" \eta is the generalized Verma module for the universal enveloping algebra \UEA(\gLie), associated with the data consisting of the pairs (H, \eta(H)) \in \UEA(\gLie) \times \bbk for H \in \gLie^0 and pairs (E,0) \in \UEA(\gLie) \times \bbk for E \in \gLie^+. The Verma module is (by a mild abuse of notation) still denoted by \Verma^{\eta}.