Adjoining the degree derivation

Let $𝔤$ be a $𝔄$ -Graded Lie algebra over $𝕂$ with $𝔄 \leq 𝕂 +$ a submonoid, so that we may define the degree derivation $𝑑 \in D (𝔤)$ . Then by adjoining the derivation $𝑑$ to $𝔤$ one gets a unique graded structure on $𝔤 ⋊ 𝕂 𝑑$ such that $a d 𝑑$ is the degree operator, #m/def/lie whence $d e g 𝑑 = 0$ .

Modules

Let $𝔤$ be a $𝔄$ -Graded Lie algebra over $𝕂$ with $𝔄 \leq 𝕂 +$ a submonoid and $𝑉$ be a graded module* over $𝔤$ . Then $𝑉$ is also a graded module over $𝔤 ⋊ 𝕂 𝑑$ where $𝑑$ acts as the Degree operator on $𝑉$ , i.e. $𝑑 \cdot 𝑣 = 𝛼 𝑣$ for $𝑣 \in 𝑉 𝛼$ . #m/thm/lie

Proof

#missing/proof

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