Supersymmetric extension of universal enveloping vertex algebras

In this paper, we study the construction of the supersymmetric extensions of vertex algebras. In particular, for $N=n\in {\mathbb{Z}}_{+}$, we show the universal enveloping $N=n$ supersymmetric (SUSY) vertex algebra of an $N=n$ SUSY Lie conformal algebra can be extended to an $N=n^{\mathrm{\prime }}>n$ SUSY vertex algebra.