On Indecomposable Non-Simple $\mathbb{N}$-graded Vertex Algebras

Abstract

In this paper, we study an impact of Leibniz algebras on the algebraic structure of $\mathbb{N}$-graded vertex algebras. We provide easy ways to characterize indecomposable non-simple $\mathbb{N}$-graded vertex algebras ${\oplus }_{n=0}^{\infty }{V}_{(n)}$ such that $\dim V_{(0)}\ge 2$. Also, we examine the algebraic structure of $\mathbb{N}$-graded vertex algebras $V={\oplus }_{n=0}^{\infty }{V}_{(n)}$ such that $\dim V_{(0)}\ge 2$ and $V_{(1)}$ is a (semi)simple Leibniz algebra that has $s{l}_2$ 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 $s{l}_2$ as their Levi factor.

25 pages

References in corpus (3)

• Modular invariance of trace functions in orbifold theory
• Chiral Poincaré duality
• Vertex Algebroids II