Abstract

In this paper, we study an impact of Leibniz algebras on the algebraic structure of N-graded vertex algebras. We provide easy ways to characterize indecomposable non-simple N-graded vertex algebras ⊕<inf>n=0</inf> ^∞V<inf>(n)</inf> such that dim⁡V<inf>(0)</inf>≥2. Also, we examine the algebraic structure of N-graded vertex algebras V=⊕<inf>n=0</inf> ^∞V<inf>(n)</inf> such that dim⁡V<inf>(0)</inf>≥2 and V<inf>(1)</inf> is a (semi)simple Leibniz algebra that has sl<inf>2</inf> 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<inf>2</inf> as their Levi factor.