[[Lie algebras MOC]]
# Triangular Lie algebra

Let $\mathfrak{g}$ be a [[Lie algebra]] over $\mathbb{K}$.
A **triangular decomposition** of $\mathfrak{g}$ is a triple of [[Lie subalgebra|subalgebras]] $\mathfrak{n}^\pm, \mathfrak{ h} \leq \mathfrak{g}$
such that
$$
\begin{align*}
\mathfrak{g} = \mathfrak{n}^{-} \oplus \mathfrak{h} \oplus \mathfrak{n}^+
\end{align*}
$$
where $\mathfrak{h}$ is [[Abelian Lie algebra|abelian]] and $[\mathfrak{h}, \mathfrak{n}^\pm] \sube \mathfrak{n}^\pm$.[^1988] #m/def/lie 
A Lie algebra with such a decomposition is called **triangular**.
This may be viewed as a generalization of a [[Heisenberg algebra]].

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

## Properties

- [[Triangular module]]
- [[Contravariant form on a triangular module]]

## Examples

- [[Special linear Lie algebra]]

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#state/tidy  | #lang/en | #SemBr