Neveu-Schwarz and operators algebras I: Vertex operators superalgebras

This paper is the first of a series giving a self-contained way from the Neveu-Schwarz algebra to a new series of irreducible subfactors. Here we present an elementary, progressive and self-contained approch to vertex operator superalgebra. We then build such a structure from the loop algebra $Lg$ of any simple finite dimensional Lie algebra $g$. The Neveu-Schwarz algebra $Vi{r}_{1\mathrm{/}2}$ emerges naturally on. As application, we obtain a unitary action of $Vi{r}_{1\mathrm{/}2}$ on the unitary discrete series of $Lg$.