On Indecomposable Vertex Algebras associated with Vertex Algebroids

TL;DR

This paper constructs a specific class of indecomposable, non-simple, $C_2$-cofinite vertex algebras from vertex algebroids with Levi factor $sl_2$, revealing their module structure and properties.

Contribution

It introduces a method to build indecomposable non-simple vertex algebras from vertex algebroids with Levi $sl_2$, expanding understanding of their structure and modules.

Findings

The constructed vertex algebra is $C_2$-cofinite.

It has exactly two irreducible modules.

The algebra is indecomposable and non-simple.

Abstract

Let $A$ be a finite dimensional unital commutative associative algebra and let $B$ be a finite dimensional vertex $A$-algebroid such that its Levi factor is isomorphic to $sl_2$. Under suitable conditions, we construct an indecomposable non-simple $\mathbb{N}$-graded vertex algebra $\overline{V_B}$ from the $\mathbb{N}$-graded vertex algebra $V_B$ associated with the vertex $A$-algebroid $B$. We show that this indecomposable non-simple $\mathbb{N}$-graded vertex algebra $\overline{V_B}$ is $C_2$-cofinite and has only two irreducible modules.