Fusion and (non)-rigidity of Virasoro Kac modules in logarithmic minimal models at $(p,q)$-central charge

Let $\mathcal{O}_c$ be the category of finite-length modules for the Virasoro Lie algebra at central charge $c$ whose composition factors are irreducible quotients of reducible Verma modules. For any $c\in\mathbb{C}$, this category admits the vertex algebraic braided tensor category structure of Huang, Lepowsky, and Zhang. Here, we begin the detailed study of $\mathcal{O}_{c_{p,q}}$ where $c_{p,q} = 1-\frac{6(p-q)^2}{pq}$ for relatively prime integers $p, q \geq 2$; in conformal field theory, $\mathcal{O}_{c_{p,q}}$ corresponds to a logarithmic extension of the central charge $c_{p,q}$ Virasoro minimal model. We particularly focus on the Virasoro Kac modules $\mathcal{K}_{r,s}$, $r,s\in\mathbb{Z}_{\geq 1}$, in $\mathcal{O}_{c_{p,q}}$ defined by Morin-Duchesne, Rasmussen, and Ridout, which are finitely-generated submodules of Feigin-Fuchs modules for the Virasoro algebra. We prove that $\mathcal{K}_{r,s}$ is rigid and self-dual when $1\leq r\leq p$ and $1\leq s\leq q$, but that not all $\mathcal{K}_{r,s}$ are rigid when $r>p$ or $s>q$. That is, $\mathcal{O}_{c_{p,q}}$ is not a rigid tensor category. We also show that all Kac modules and all simple modules in $\mathcal{O}_{c_{p,q}}$ are homomorphic images of repeated tensor products of $\mathcal{K}_{1,2}$ and $\mathcal{K}_{2,1}$, and we determine completely how $\mathcal{K}_{1,2}$ and $\mathcal{K}_{2,1}$ tensor with Kac modules and simple modules in $\mathcal{O}_{c_{p,q}}$. In the process, we prove some fusion rule conjectures of Morin-Duchesne, Rasmussen, and Ridout.