Degree operator

Adjoining the degree derivation

Let 𝔤 be a 𝔄-Graded Lie algebra over 𝕂 with 𝔄 ≤𝕂+ a submonoid, so that we may define the degree derivation 𝑑 ∈D(𝔤). Then by adjoining the derivation 𝑑 to 𝔤 one gets a unique graded structure on 𝔤 ⋊𝕂𝑑 such that ad𝑑 is the degree operator, lie whence deg⁡𝑑 =0.

Modules

Let 𝔤 be a 𝔄-Graded Lie algebra over 𝕂 with 𝔄 ≤𝕂+ a submonoid and 𝑉 be a graded module* over 𝔤. Then 𝑉 is also a graded module over 𝔤 ⋊𝕂𝑑 where 𝑑 acts as the Degree operator on 𝑉, i.e. 𝑑 ⋅𝑣 =𝛼𝑣 for 𝑣 ∈𝑉𝛼. lie

Proof

proof

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