[[Triangular Lie algebra]]
# Triangular module

Let $\mathfrak{g} = \mathfrak{n}^{-} \oplus \mathfrak{h} \oplus \mathfrak{n}^+$ be a [[triangular Lie algebra]] over $\mathbb{K}$
and $\lambda : \mathfrak{h} \to \mathbb{K}$ be a [[linear form]].
The **triangular module** $M(\lambda)$ is the [[Induced module|induced]] $\mathfrak{g}$-[[Module over a Lie algebra|module]] #m/def/lie
$$
\begin{align*}
M(\lambda) = \Ind_{\mathfrak{h} \oplus \mathfrak{n}^+}^\mathfrak{g} \mathbb{K}_{\lambda}
\end{align*}
$$
where the [[vacuum space]] $\mathbb{K}_{\lambda} = \mathbb{K}v_{\lambda}$ is the nonzero $(\mathfrak{h}\oplus \mathfrak{n}^+)$-module defined by[^1988]
$$
\begin{align*}
\mathfrak{n}^+ \odot  v_{\lambda} &= 0 & h \odot  v_{\lambda} = \lambda(h)v_{\lambda}
\end{align*}
$$
for $h \in \mathfrak{h}$.
This is a direct generalization of the [[Heisenberg module]] $M(k)$.

  [^1988]: 1988\. [[Sources/@frenkelVertexOperatorAlgebras1988|Vertex operator algebras and the Monster]], §1.8, p. 26

## Properties

- [[Contravariant form on a triangular module]]

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