Graded vector space

Tensor product of graded vector spaces

Let 𝑉,𝑊 be 𝔄-graded vector spaces over 𝕂 for some monoid (𝔄, +). The tensor product 𝑉 ⊗𝑊 is the Tensor product of vector spaces over a field with the unique 𝔄-gradation specified by linalg

𝑉𝛼⊗𝑊𝛽≤(𝑉⊗𝑊)𝛼+𝛽

for all 𝛼,𝛽 ∈𝔄. This extends to any finite number of factors inductively.

Proof of uniqueness

proof

Properties

1. The Degree operator is given by 𝑑𝑉⊗𝑊 =𝑑𝑉 ⊗1 +1 ⊗𝑑𝑊.

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