Vertex algebroids and Conformal vertex algebras associated with simple Leibniz algebras

TL;DR

This paper explores the structure of vertex algebroids derived from simple Leibniz algebras and constructs associated non-simple vertex algebras, classifying their modules and analyzing conformal vectors.

Contribution

It introduces a new connection between simple Leibniz algebras and vertex algebroids, leading to the construction and classification of novel vertex algebras.

Findings

Constructed indecomposable non-simple $C_2$-cofinite $ $-graded vertex algebras

Classified $ $-graded irreducible modules of these vertex algebras

Analyzed conformal vectors within the constructed vertex algebras

Abstract

We first investigate the algebraic structure of vertex algebroids $B$ when $B$ are simple Leibniz algebras. Next, we use these vertex algebroids $B$ to construct indecomposable non-simple $C_2$-cofinite $\mathbb{N}$-graded vertex algebras $\overline{V_B}$. In addition, we classify $\mathbb{N}$-graded irreducible $\overline{V_B}$-modules and examine conformal vectors of these $\mathbb{N}$-graded vertex algebras $\overline{V_B}$.