Freely generated vertex algebras and non-linear Lie conformal algebras

TL;DR

This paper introduces non-linear Lie conformal superalgebras and establishes a PBW theorem for their universal enveloping vertex algebras, linking freely generated vertex algebras to these structures.

Contribution

It defines non-linear Lie conformal superalgebras and proves a PBW theorem, creating a new framework for classifying graded vertex algebras.

Findings

Proved a PBW theorem for non-linear Lie conformal superalgebras.

Showed the correspondence between graded freely generated vertex algebras and their enveloping algebras.

Established a foundation for classifying finitely generated simple graded vertex algebras.

Abstract

We introduce the notion of a non--linear Lie conformal superalgebra and prove a PBW theorem for its universal enveloping vertex algebra. We also show that conversely any graded freely generated vertex algebra is the universal enveloping algebra of a non--linear Lie conformal superalgebra. This correspondence will be applied in the subsequent work to the problem of classification of finitely generated simple graded vertex algebras.