Degree operator

Let $𝑉$ be an $𝑆$ -Graded vector space over $𝕂$ where $𝑆 \leq 𝕂 +$ is a submonoid of the additive group. Then the degree operator $𝑑 𝑉 : 𝑉 \to 𝑉$ is defined by #m/def/linalg

𝑑 𝑉 𝑣 = 𝛼 𝑣

for any $𝑣 \in 𝑉 𝛼$ and $𝛼 \in 𝑆$ .

On a graded algebra

If $(𝐴, \cdot)$ is an $𝑀$ -Graded algebra over $𝕂$ where $𝑀 \leq 𝕂 +$ is a submonoid of the additive group, the degree operator $𝑑 : 𝐴 \to 𝐴$ is a derivation, #m/thm/falg called the degree derivation.

Proof

Note that for homogenous elements $𝑎 \in 𝐴 𝛼$ and $𝑏 \in 𝐵 𝛽$ we have

𝑑 (𝑎 \cdot 𝑏) = (𝛼 + 𝛽) 𝑎 \cdot 𝑏 = 𝛼 𝑎 \cdot 𝑏 + 𝑎 \cdot 𝛽 𝑏 = 𝑑 (𝑎) \cdot 𝑏 + 𝑎 \cdot 𝑑 (𝑏)

so by linearity $𝑑$ is a derivation.

In the case $𝐴$ is a $𝑀$ -graded Lie algebra, see adjoining the degree derivation.

Properties

Let $𝑓 : 𝑉 \to 𝑊$ be a linear map between $𝑆$ -graded vector spaces over $𝕂$ where $𝑆 \leq 𝕂 +$

$𝑓$ is graded iff $[𝑑, 𝑓] = 𝑑 𝑊 𝑓 - 𝑓 𝑑 𝑉 = 0$
$𝑓$ is homogenous of degree $𝛽 \in 𝑆$ iff $[𝑑, 𝑓] = 𝑑 𝑊 𝑓 - 𝑓 𝑑 𝑉 = 𝛽 𝑓$

Proof

#missing/proof

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