Formal sums over a vector space

Degree operator on formal sums over a vector space

Let 𝑉 be a vector space over 𝕂 and 𝑉{𝑧} denote formal sums over 𝑉. We define the 𝑧-degree operator¹ #m/def/fcalc

where 𝑑𝑑𝑧 is the Formal derivative. This is not strictly a degree operator, as 𝑉{𝑧} is not a graded vector space, however its subspaces 𝑉[𝑧] and 𝑉[𝑧,𝑧−1] are.

Properties

Over a graded vector space

Now taking 𝑉 to be 𝕂-graded with degree operator 𝑑 ∈End⁡𝑉, and letting

we have

With the formal Dirac delta

Let 𝛿(𝑧) ∈𝕂[[𝑧,𝑧−1]] denote the Formal delta. Then it follows from Properties that for any 𝑣(𝑧) ∈𝑣[𝑧,𝑧−1] and 𝑎 ∈𝕂×

and that for any 𝑋(𝑧1,𝑧2) ∈(End⁡𝑉)[[𝑧1,𝑧−11,𝑧2,𝑧−12]] such that lim𝑧1→𝑧2𝑋(𝑧1,𝑧2) exists and 𝑎 ∈𝕂×

where 𝐷1 =𝐷𝑧1 and 𝐷2 =𝐷𝑧2.

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