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Let $mathcal{O}_c$ be the category of finite-length central-charge-$c$ modules for the Virasoro Lie algebra whose composition factors are irreducible quotients of reducible Verma modules. Recently, it has been shown that $mathcal{O}_c$ admits vertex algebraic tensor category structure for any $cinmathbb{C}$. Here, we determine the structure of this tensor category when $c=13-6p-6p^{-1}$ for an integer $p>1$. For such $c$, we prove that $mathcal{O}_{c}$ is rigid, and we construct projective covers of irreducible modules in a natural tensor subcategory $mathcal{O}_{c}^0$. We then compute all tensor products involving irreducible modules and their projective covers. Using these tensor product formulas, we show that $mathcal{O}_c$ has a semisimplification which, as an abelian category, is the Deligne product of two tensor subcategories that are tensor equivalent to the Kazhdan-Lusztig categories for affine $mathfrak{sl}_2$ at levels $-2+p^{pm 1}$. Next, as a straightforward consequence of the braided tensor category structure on $mathcal{O}_c$ together with the theory of vertex operator algebra extensions, we rederive known results for triplet vertex operator algebras $mathcal{W}(p)$, including rigidity, fusion rules, and construction of projective covers. Finally, we prove a recent conjecture of Negron that $mathcal{O}_c^0$ is braided tensor equivalent to the $PSL(2,mathbb{C})$-equivariantization of the category of $mathcal{W}(p)$-modules.
We provide a ribbon tensor equivalence between the representation category of small quantum SL(2), at parameter q=exp($pi$ i/p), and the representation category of the triplet vertex operator algebra at integral parameter p>1. We provide similar quan
We show that direct limit completions of vertex tensor categories inherit vertex and braided tensor category structures, under conditions that hold for example for all known Virasoro and affine Lie algebra tensor categories. A consequence is that the
We compute the group of braided tensor autoequivalences and the Brauer-Picard group of the representation category of the small quantum group $mathfrak{u}_q(mathfrak{g})$, where $q$ is a root of unity.
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Let $Vsubseteq A$ be a conformal inclusion of vertex operator algebras and let $mathcal{C}$ be a category of grading-restricted generalized $V$-modules that admits the vertex algebraic braided tensor category structure of Huang-Lepowsky-Zhang. We giv