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Let V be a simple VOA and consider a representation category of V that is avertex tensor category in the sense of Huang-Lepowsky. In particular, thiscategory is a braided tensor category. Let J be an object in this category thatis a simple current of order two of either integer or half-integer conformaldimension. We prove that $V\oplus J$ is either a VOA or a super VOA. If therepresentation category of V is in addition ribbon, then the categoricaldimension of J decides this parity question. Combining with Carnahan's work, weextend this result to simple currents of arbitrary order. Our next result is asimple sufficient criterion for lifting indecomposable objects that onlydepends on conformal dimensions. Several examples of simple current extensionsthat are $C_2$-cofinite and non-rational are then given and induced moduleslisted.

Simple current extensions beyond semi-simplicity