Abstract
The main goals for this paper are i) to study of the algebraic structure of N-graded vertex algebras VB associated with vertex A-algebroids B when B are cyclic non-Lie left Leibniz algebras, and ii) to explore relations between the vertex algebras VB and the rank one Heisenberg vertex operator algebra. To achieve these goals, we first classify vertex A-algebroids B associated with 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 VB. Finally, we use the algebraic structure of the unital commutative associative algebras A that we found to study relations between certain types of the vertex algebras VB and the vertex operator algebra associated to the rank one Heisenberg algebra.
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