Finite induction functor for vertex operator algebras

In this paper, we introduce a new induction functor $\mathrm{Ind}^V_U$ between module categories corresponding to an embedding of vertex operator algebras (VOAs) $U \hookrightarrow V$. This induction functor is essentially defined at the level of the finite (Zhu) algebras, which we call the \emph{finite induction functor}. Under suitable conditions on $U$ and $V$, we prove that this functor satisfies the usual properties of induction functors, such as Frobenius reciprocity, functorial property for compositions, and an analogue of Artin's induction theorem for certain associated characters. To better understand the effect of this functor, we explicitly determine the finite induction of irreducible modules for standard subVOAs of the rank-one lattice/affine VOA $V_{A_1}$, as well as the finite induction of irreducible modules over a parabolic-type subVOA $V_P$ of the rank-two lattice/affine VOA $V_{A_2}$.