VOA conditions for being a connected Frobenius algebra object in Rep(V)

Identify and characterize conditions under which a vertex operator algebra V is a connected Frobenius algebra object in its representation category Rep(V).

Background

From the categorical perspective on VOA extensions, a VOA extension A of V yields a commutative, haploid algebra in Rep(V) with a trivial twist. However, it is not fully established that such extensions always realize a Frobenius algebra structure.

Clarifying when VOAs become connected Frobenius algebra objects in Rep(V) would enable applications of the paper’s finiteness theorems to VOA settings and deepen the link between tensor-categorical structures and vertex operator algebras.