[[Semidirect product]]
# Adjoining a derivation

Let $\mathfrak{g}$ be a [[Lie algebra]] over $\mathbb{K}$ and $d$ be a [[Derivation on an algebra|derivation]] on $\mathfrak{g}$.
Then one may **adjoin the derivation** $d$ to $\mathfrak{g}$ by taking the [[Semidirect product of Lie algebras|semidirect product]] #m/def/lie 
$$
\begin{align*}
\mathfrak{g} \rtimes  \mathbb{K}d
\end{align*}
$$
where $\mathbb{K}d$ is the free vector space generated by $d$ with the unique ([[Abelian Lie algebra|abelian]]) bracket.[^1988]

  [^1988]: 1988\. [[Sources/@frenkelVertexOperatorAlgebras1988|Vertex operator algebras and the Monster]], §1.3, p. 8

## Special cases

- [[Adjoining the degree derivation]]

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