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 $p,q\ge 2$; 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 ${\mathcal{K}}_{r,s}$, $r,s\in {\mathbb{Z}}_{\ge 1}$, 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 $1\le r\le p$ and $1\le s\le q$, 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.