Simple current extensions beyond semi-simplicity

Understand

Let V be a simple VOA and consider a representation category of V that is a vertex tensor category in the sense of Huang-Lepowsky.

• In particular, this category is a braided tensor category.
• Let J be an object in this category that is a simple current of order two of either integer or half-integer conformal dimension.
• We prove that $V\oplus J$ is either a VOA or a super VOA.