Supersymmetric extension of universal enveloping vertex algebras

In this paper, we study the construction of the supersymmetric extensions of vertex algebras. In particular, for $N=n\in Z_{+}$, we show that the universal enveloping $N=n$ SUSY vertex algebra of an $N=n$ SUSY Lie conformal algebra can be extended to an $N=n^{\prime }>n$ SUSY vertex algebra. Additionally, we investigate various superconformal vectors which induce the same SUSY structure but distinct conformal weight decompositions of a vertex algebra.