Formal delta

The formal delta over a field $𝕂$ is the Laurent series¹ #m/def/fcalc

𝛿 (𝑧) = \sum 𝑛 \in ℤ 𝑧 𝑛 \in 𝕂 [[𝑧, 𝑧 - 1]]

given by the Fourier series expansion of the Dirac delta.

Properties

Let $𝑉$ be a vector space over $𝕂$ . Let $𝑣 (𝑧) \in 𝑉 [𝑧, 𝑧 - 1]$ and $𝑎 \in 𝕂 \times$ . Finally let $𝑝 (𝑧) \in 𝕂 [𝑧, 𝑧 - 1]$ and $𝑇 = 𝑇 𝑝 (𝑧) = 𝑝 (𝑧) 𝑑 𝑑 𝑧$ . Then in $𝑉 {𝑧}$

$𝑣 (𝑧) 𝛿 (𝑎 𝑧) = 𝑣 (𝑎 - 1) 𝛿 (𝑎 𝑧)$
$𝑣 (𝑧) 𝑑 𝑑 𝑧 [𝛿 (𝑎 𝑧)] = 𝑣 (𝑎 - 1) 𝑑 𝑑 𝑧 [𝛿 (𝑎 𝑧)] - 𝑣' (𝑎 - 1) 𝛿 (𝑎 𝑧)$
$𝑣 (𝑧) 𝑇 [𝛿 (𝑎 𝑧)] = 𝑣 (𝑎 - 1) 𝑇 [𝛿 (𝑎 𝑧)] - (𝑇 𝑣) (𝑎 - 1) 𝛿 (𝑎 𝑧)$

Let $𝑋 (𝑧 1, 𝑧 2) \in (E n d 𝑉) [[𝑧 1, 𝑧 - 1 1, 𝑧 2, 𝑧 - 1 2]]$ such that $l i m 𝑧 1 \to 𝑧 2 𝑋 (𝑧 1, 𝑧 2)$ exists and $𝑎 \in 𝕂 \times$ . Finally let $𝑝 (𝑧 1, 𝑧 2) \in 𝕂 [𝑧 1, 𝑧 - 1 1, 𝑧 2, 𝑧 - 1 2]$ , $𝑇 1 = 𝑝 (𝑧 1, 𝑧 2) 𝜕 𝜕 𝑧 1$ , and $𝑇 2 = 𝑝 (𝑧 1, 𝑧 2) 𝜕 𝜕 𝑧 2$ . Then in $(E n d 𝑉) {𝑧 1, 𝑧 2}$

$𝑋 (𝑧 1, 𝑧 2) 𝛿 (𝑎 𝑧 1 / 𝑧 2) = 𝑋 (𝑎 - 1 𝑧 2, 𝑧 2) 𝛿 (𝑎 𝑧 1 / 𝑧 2) = 𝑋 (𝑧 1, 𝑎 𝑧 1) 𝛿 (𝑎 𝑧 1 / 𝑧 2)$
$𝑋 (𝑧 1, 𝑧 2) 𝜕 𝜕 𝑧 1 [𝛿 (𝑎 𝑧 1 / 𝑧 2)] = 𝑋 (𝑎 - 1 𝑧 2, 𝑧 2) 𝜕 𝜕 𝑧 1 [𝛿 (𝑎 𝑧 1 / 𝑧 2)] - (𝜕 𝑋 𝜕 𝑧 1) (𝑎 - 1 𝑧 2, 𝑧 2) 𝛿 (𝑎 𝑧 1 / 𝑧 2) 𝑋 (𝑧 1, 𝑧 2) 𝜕 𝜕 𝑧 2 [𝛿 (𝑎 𝑧 1 / 𝑧 2)] = 𝑋 (𝑧 1, 𝑎 𝑧 1) 𝜕 𝜕 𝑧 2 [𝛿 (𝑎 𝑧 1 / 𝑧 2)] - (𝜕 𝑋 𝜕 𝑧 2) (𝑧 1, 𝑎 𝑧 1) 𝛿 (𝑎 𝑧 1 / 𝑧 2)$
$𝑋 (𝑧 1, 𝑧 2) 𝑇 1 [𝛿 (𝑎 𝑧 1 / 𝑧 2)] = 𝑋 (𝑎 - 1 𝑧 2, 𝑧 2) 𝑇 1 [𝛿 (𝑎 𝑧 1 / 𝑧 2)] - (𝑇 1 𝑋) (𝑎 - 1 𝑧 2, 𝑧 2) 𝛿 (𝑎 𝑧 1 / 𝑧 2) 𝑋 (𝑧 1, 𝑧 2) 𝑇 2 [𝛿 (𝑎 𝑧 1 / 𝑧 2)] = 𝑋 (𝑧 1, 𝑎 𝑧 1) 𝑇 2 [𝛿 (𝑎 𝑧 1 / 𝑧 2)] - (𝑇 2 𝑋) (𝑧 1, 𝑎 𝑧 1) 𝛿 (𝑎 𝑧 1 / 𝑧 2)$

Note these fail for non-integer powers.

Proof of 1–6

First we prove ^P1. Consider the special case $𝑣 (𝑧) = 𝑣 𝑛 𝑧 𝑛$ . Then

𝑣 (𝑧) 𝛿 (𝑎 𝑧) = (𝑣 𝑛 𝑧 𝑛) (\sum 𝑘 \in ℤ 𝑎 𝑘 𝑧 𝑘) = \sum 𝑘 \in ℤ 𝑎 𝑘 𝑣 𝑛 𝑧 𝑘 + 𝑛 = \sum 𝑘 \in ℤ 𝑎 𝑘 - 𝑛 𝑣 𝑛 𝑧 𝑘 = (𝑎 - 𝑛 𝑣 𝑛) (\sum 𝑘 \in ℤ 𝑎 𝑘 𝑧 𝑘) = 𝑣 (𝑎 - 1) 𝛿 (𝑎 𝑧)

whence follows ^P1 by linearity. The proof of ^PA is similar. Let $𝑋 (𝑧 1, 𝑧 2) = \sum 𝑚, 𝑛 \in ℤ 𝑧 𝑚 1 𝑧 𝑛 2$ . Then

𝑋 (𝑧 1, 𝑧 2) 𝛿 (𝑎 𝑧 1 / 𝑧 2) = (\sum 𝑚, 𝑛 \in ℤ 𝑥 (𝑚, 𝑛) 𝑧 𝑚 1 𝑧 𝑛 2) (\sum 𝑘 \in ℤ 𝑎 𝑘 𝑧 𝑘 1 𝑧 - 𝑘 2) = \sum 𝑚, 𝑛, 𝑘 \in ℤ 𝑎 𝑘 𝑥 (𝑚, 𝑛) 𝑧 𝑚 + 𝑘 1 𝑧 𝑛 - 𝑘 2 = \sum 𝑚, 𝑛, 𝑘 \in ℤ 𝑎 𝑘 - 𝑚 𝑥 (𝑚, 𝑛) 𝑧 𝑘 1 𝑧 𝑚 + 𝑛 - 𝑘 2 = 𝑋 (𝑎 - 1 𝑧 2, 𝑧 2) 𝛿 (𝑎 𝑧 1 / 𝑧 2)

Then ^P3 and ^PC follow by taking appropriate derivatives, and ^P2 and ^PB are special cases.

#state/develop | #lang/en | #SemBr

1988. Vertex operator algebras and the Monster, §2.1–§2.2, p. 52ff ↩

Formal delta

Properties

Footnotes