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

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.