On vertex Leibniz algebras

TL;DR

This paper introduces the concept of vertex Leibniz algebras, extending vertex algebras without vacuum, and explores their embeddings into vertex algebras, highlighting conditions related to faithful modules and Lie algebra structures.

Contribution

It defines vertex Leibniz algebras, relates them to vertex algebras without vacuum, and characterizes their embeddability based on faithful modules and Lie algebra properties.

Findings

Every vertex algebra without vacuum extends to a vertex algebra.

A vertex Leibniz algebra embeds into a vertex algebra iff it has a faithful module.

The constructed vertex Leibniz algebra V_g embeds into a vertex algebra iff g is a Lie algebra.

Abstract

In this paper, we study a notion of what we call vertex Leibniz algebra. This notion naturally extends that of vertex algebra without vacuum, which was previously introduced by Huang and Lepowsky. We show that every vertex algebra without vacuum can be naturally extended to a vertex algebra. On the other hand, we show that a vertex Leibniz algebra can be embedded into a vertex algebra if and only if it admits a faithful module. To each vertex Leibniz algebra we associate a vertex algebra without vacuum which is universal to the forgetful functor. Furthermore, from any Leibniz algebra $\g$ we construct a vertex Leibniz algebra $V_{\g}$ and show that $V_{\g}$ can be embedded into a vertex algebra if and only if $\g$ is a Lie algebra.