On $\mathbb{N}$-graded vertex algebras associated with cyclic Leibniz algebras with small dimensions

TL;DR

This paper classifies and constructs $ $-graded vertex algebras from cyclic Leibniz algebras, exploring their structure and relation to the rank one Heisenberg vertex operator algebra, revealing new classes of indecomposable non-simple algebras.

Contribution

It provides a classification of vertex $A$-algebroids from cyclic Leibniz algebras and constructs associated indecomposable non-simple vertex algebras, linking them to the Heisenberg algebra.

Findings

Classified vertex $A$-algebroids for cyclic Leibniz algebras.

Constructed a family of indecomposable non-simple vertex algebras.

Established relations between these vertex algebras and the rank one Heisenberg algebra.

Abstract

The main goals for this paper is i) to study of an algebraic structure of $\mathbb{N}$-graded vertex algebras $V_B$ associated to vertex $A$-algebroids $B$ when $B$ are cyclic non-Lie left Leibniz algebras, and ii) to explore relations between the vertex algebras $V_B$ and the rank one Heisenberg vertex operator algebra. To achieve these goals, we first classify vertex $A$-algebroids $B$ associated to given cyclic non-Lie left Leibniz algebras $B$. Next, we use the constructed vertex $A$-algebroids $B$ to create a family of indecomposable non-simple vertex algebras $V_B$. Finally, we use the algebraic structure of the unital commutative associative algebras $A$ that we found to study relations between a certain type of the vertex algebras $V_B$ and the vertex operator algebra associated to a rank one Heisenberg algebra.