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The $\mathcal{B}_p$-algebras are a family of vertex operator algebrasparameterized by $p\in \mathbb Z_{\geq 2}$. They are important examples oflogarithmic CFTs and appear as chiral algebras of type $(A_1, A_{2p-3})$Argyres-Douglas theories. The first member of this series, the$\mathcal{B}_2$-algebra, are the well-known symplectic bosons also often calledthe $\beta\gamma$ vertex operator algebra.  We study categories related to the $\mathcal{B}_p$ vertex operator algebrasusing their conjectural relation to unrolled restricted quantum groups of$\mathfrak{sl}_2$. These categories are braided, rigid and non semi-simpletensor categories. We list their simple and projective objects, their tensorproducts and their Hopf links. The latter are successfully compared to modulardata of characters thus confirming a proposed Verlinde formula of David Ridoutand the second author.

Braided Tensor Categories related to $\mathcal{B}_p$ Vertex Algebras