Graded module

Vacuum space

Let be a -graded -monoid1 and be a graded -module with the action denoted by . A nonzero vector is called a vacuum vector iff . #m/def/ralg The vacuum space consists of all vacuum vectors and zero

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

Proof

Let be a vacuum vector. Then for any and

so for all and .

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

Footnotes

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

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