Graded vector space

Degree operator

Let 𝑉 be an 𝑆-Graded vector space over 𝕂 where 𝑆 ≤𝕂+ is a submonoid of the additive group. Then the degree operator 𝑑𝑉 :𝑉 →𝑉 is defined by linalg

𝑑𝑉𝑣=𝛼𝑣

for any 𝑣 ∈𝑉𝛼 and 𝛼 ∈𝑆.

On a graded algebra

If (𝐴, ⋅) is an 𝑀-Graded algebra over 𝕂 where 𝑀 ≤𝕂+ is a submonoid of the additive group, the degree operator 𝑑 :𝐴 →𝐴 is a derivation, falg called the degree derivation.

Proof

Note that for homogenous elements 𝑎 ∈𝐴𝛼 and 𝑏 ∈𝐵𝛽 we have

𝑑(𝑎⋅𝑏)=(𝛼+𝛽)𝑎⋅𝑏=𝛼𝑎⋅𝑏+𝑎⋅𝛽𝑏=𝑑(𝑎)⋅𝑏+𝑎⋅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 →𝑊 be a linear map between 𝑆-graded vector spaces over 𝕂 where 𝑆 ≤𝕂+

1. 𝑓 is ^graded iff [𝑑,𝑓] =𝑑𝑊𝑓 −𝑓𝑑𝑉 =0
2. 𝑓 is ^homogenous of degree 𝛽 ∈𝑆 iff [𝑑,𝑓] =𝑑𝑊𝑓 −𝑓𝑑𝑉 =𝛽𝑓

Proof

proof

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