On semisimplicity of module categories for finite non-zero index vertex operator subalgebras

Let $V\subseteq A$ be a conformal inclusion of vertex operator algebras and let $\mathcal{C}$ category grading-restricted generalized $V$-modules that admits the algebraic braided tensor structure Huang-Lepowsky-Zhang. We give conditions under which inherits semisimplicity from $A$-modules in $\mathcal{C}$, vice versa. The most important condition is $A$ rigid $V$-module with non-zero categorical dimension, is, we assume index $V$ as subalgebra finite non-zero. As consequence, show if strongly rational, then also rational following conditions: contains direct summand, $C_2$-cofinite modules, has dimension $V$-module. These results are algebra interpretations theorems proved for general commutative categories. generalize these to case superalgebra.