Embedding of certain vertex algebras without vacuum into vertex algebras

TL;DR

This paper demonstrates that certain vertex algebras lacking a vacuum vector can be embedded into full vertex algebras, providing a more elementary proof and establishing the canonicity of the construction.

Contribution

It introduces a new, elementary proof for embedding vacuum-free vertex algebras into vertex algebras, extending previous results and showing the canonicity of the embeddings.

Findings

Embedding of vacuum-free vertex algebras into vertex algebras is possible.

The proof is more elementary than previous methods.

The constructions are canonical.

Abstract

We show that certain vertex algebras without vacuum vector may be embedded into vertex algebras. The result is a partial analogue of the simple classical fact that any rng can be embedded into a ring. A one-line proof of the case of a vacuum-free vertex algebra (whose vertex operator map is by definition injective) appeared in Robinson (2010) using a powerful result from the representation theory of vertex algebras as algebras of mutually local weak vertex operators. Here we present a more elementary proof of a somewhat more general case. We also show that our constructions are canonical.