[[Degree operator]]
# Adjoining the degree derivation

Let $\mathfrak{g}$ be a $\mathfrak{A}$-[[Graded Lie algebra]] over $\mathbb{K}$ with $\mathfrak{A} \leq \mathbb{K}^+$ a [[submonoid]],
so that we may define the [[Degree operator|degree derivation]] $d \in \mathcal{D}(\mathfrak{g})$.
Then by [[Adjoining a derivation|adjoining the derivation]] $d$ to $\mathfrak{g}$
one gets a unique [[Graded algebra|graded]] structure on $\mathfrak{g} \rtimes \mathbb{K}d$ such that $\ad_{d}$ is the degree operator,  #m/def/lie
whence $\deg d = 0$.

## Modules

Let $\mathfrak{g}$ be a $\mathfrak{A}$-[[Graded Lie algebra]] over $\mathbb{K}$ with $\mathfrak{A} \leq \mathbb{K}^+$ a [[submonoid]]
and $V$ be a [[Graded module|graded]] [[Module over a Lie algebra|module* over]] $\mathfrak{g}$.
Then $V$ is also a graded module over $\mathfrak{g} \rtimes \mathbb{K}d$ where $d$ acts as the [[Degree operator]] on $V$,
i.e. $d \cdot v = \alpha v$ for $v \in V_{\alpha}$. #m/thm/lie 

> [!missing]- Proof
> #missing/proof


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