# On Indecomposable Non-Simple $\mathbb{N}$-graded Vertex Algebras

## TL;DR

This paper investigates the structure of certain $
$-graded vertex algebras influenced by Leibniz algebras, providing classification methods and demonstrating conditions under which these algebras are indecomposable but not simple.

## Contribution

It introduces new characterizations and classifications of indecomposable non-simple $
$-graded vertex algebras with specific Leibniz algebra structures.

## Key findings

- Characterization methods for indecomposable non-simple $
$-graded vertex algebras
- Classification of vertex algebroids with $sl_2$ Levi factor
- Conditions under which these vertex algebras are indecomposable but not simple

## Abstract

In this paper, we study an impact of Leibniz algebras on the algebraic structure of $\mathbb{N}$-graded vertex algebras. We provide easy ways to characterize indecomposable non-simple $\mathbb{N}$-graded vertex algebras $\oplus_{n=0}^{\infty}V_{(n)}$ such that $\dim V_{(0)}\geq 2$. Also, we examine the algebraic structure of $\mathbb{N}$-graded vertex algebras $V=\oplus_{n=0}^{\infty}V_{(n)}$ such that $\dim~V_{(0)}\geq 2$ and $V_{(1)}$ is a (semi)simple Leibniz algebra that has $sl_2$ as its Levi factor. We show that under suitable conditions this type of vertex algebra is indecomposable but not simple. Along the way we classify vertex algebroids associated with (semi)simple Leibniz algebras that have $sl_2$ as their Levi factor.