Formal sums over a vector space

Formal sums over endomorphisms

Let 𝑉 be a vector space over 𝕂 and consider formal sums over the endomorphism ring End⁡𝑉, denoted (End⁡𝑉){𝑧}.¹ We define the following operations: fcalc

Operations

Summation

The sum of a family {𝑋𝑖(𝑧)}𝑖∈𝐼 in (End⁡𝑉){𝑧} with 𝑋𝑖(𝑧) =∑𝑛∈𝕂𝑥𝑖(𝑛)𝑧𝑛 exists iff the {𝑥𝑖(𝑛)}𝑖∈𝐼 are summable for all 𝑛 ∈𝕂, and is given by

∑𝑖∈𝐼𝑋𝑖(𝑧)=∑𝑛∈𝕂(∑𝑖∈𝐼𝑥𝑖(𝑛))𝑧𝑛

Multiplication

The product of a finite list (𝑋𝑖(𝑧))𝑟𝑖=1 in (End⁡𝑉){𝑧} exists iff for every 𝑛 ∈𝕂, the set

{𝑟∏𝑖=1𝑥𝑖(𝑛𝑖):𝑟∑𝑖=1𝑛𝑖=𝑛∧{𝑛𝑖}𝑟𝑖=1⊆𝕂}

is summable and is defined as

𝑟∏𝑖=1𝑋𝑖(𝑧)=∑𝑛∈𝕂⎛⎜ ⎜ ⎜ ⎜ ⎜⎝∑𝑛1+⋯+𝑛𝑟=𝑛𝑛1,…,𝑛𝑟∈𝕂𝑟∏𝑖=1𝑥𝑖(𝑛𝑖)⎞⎟ ⎟ ⎟ ⎟ ⎟⎠𝑧𝑛

Importantly, partitioning a product into existent subproducts and taking the product of those will give the same result, but the converse doesn’t hold: Multiplication of formal sums fails to be associative, instead satisfying partial associativity.

Counterexamples

Consider the Formal delta 𝛿(𝑧) ∈𝕂[[𝑧,𝑧−1]]. Then naïve manipulation would suggest

𝛿(𝑧)=((∞∑𝑛=0𝑧𝑛)(1−𝑧))𝛿(𝑧)!=(∞∑𝑛=0𝑧𝑛)((1−𝑧)𝛿(𝑧))=0

On the other hand, this triple product exists but contains a nonexistant subproduct

(∞∑𝑛=0𝑧𝑛)(∞∑𝑛=0𝑧−𝑛)0=0

Limits of multivariable formal sums

Let

𝑋(𝑧1,𝑧2)=∑𝑚,𝑛∈𝕂𝑥(𝑚,𝑛)𝑧𝑚1𝑧𝑛2∈(End⁡𝑉){𝑧1,𝑧2}

Then lim𝑧1→𝑧2𝑋(𝑧1,𝑧2) exists iff for every 𝑛 ∈𝕂 the family {𝑥(𝑚,𝑛 −𝑚)}𝑚∈𝕂 is summable, and is given by

lim𝑧1→𝑧2(∑𝑚,𝑛∈𝕂𝑥(𝑚,𝑛)𝑧𝑚1𝑧𝑛2)=∑𝑛∈𝕂(∑𝑚∈𝕂𝑥(𝑚,𝑛−𝑚))𝑧𝑛2

See also

• Normal ordered product

---

develop | en | SemBr

Footnotes

1. 1988. Vertex operator algebras and the Monster, §2.1, pp. 49–50 ↩