Graded module

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 ∈𝑉 is called a vacuum vector iff 𝐴+ ⊙𝑣 =0. falg The vacuum space Ω𝑉 consists of all vacuum vectors and zero

Ω𝑉={𝑣∈𝑉:𝐴+⊙𝑣=0}=⨁𝑖∈ℤΩ𝑉𝑖

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

Proof

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

𝜋𝑖(𝑥⊙𝑣)=∞∑𝑗=1𝜋𝑗(𝑥)⊙𝜋𝑖−𝑗(𝑣)=0

so 𝜋𝑗(𝑥) ⊙𝜋𝑖(𝑣) =0 for all 𝑗 >1 and 𝑖 ∈ℤ.

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Footnotes

1. Or Graded Lie algebra via the Universal enveloping algebra. ↩

2. 1988. Vertex operator algebras and the Monster, §1.7, p. 23 ↩