Vacuum space

Let $𝐴$ be a $ℤ$ -graded $𝕂$ -monoid¹ and $𝑉$ be a graded $𝐴$ -module with the action denoted by $(⊙)$ . A nonzero vector $𝑣 \in 𝑉$ is called a vacuum vector iff $𝐴 + ⊙ 𝑣 = 0$ . #m/def/ralg The vacuum space $Ω 𝑉$ consists of all vacuum vectors and zero

Ω 𝑉 = {𝑣 \in 𝑉 : 𝐴 + ⊙ 𝑣 = 0} = ⨁ 𝑖 \in ℤ Ω 𝑉 𝑖

and is a graded vector subspace, i.e. all vacuum vector are linear combinations of homogenous vacuum vectors.²

Proof

Let $𝑣 \in 𝑉$ be a vacuum vector. Then for any $𝑥 \in 𝐴 +$ and $𝑖 \in ℤ$

𝜋 𝑖 (𝑥 ⊙ 𝑣) = \infty \sum 𝑗 = 1 𝜋 𝑗 (𝑥) ⊙ 𝜋 𝑖 - 𝑗 (𝑣) = 0

so $𝜋 𝑗 (𝑥) ⊙ 𝜋 𝑖 (𝑣) = 0$ for all $𝑗 > 1$ and $𝑖 \in ℤ$ .

#state/tidy | #lang/en | #SemBr

Or Graded Lie algebra via the Universal enveloping algebra. ↩
1988. Vertex operator algebras and the Monster, §1.7, p. 23 ↩

Vacuum space

Footnotes