  The $\mathcal{B}_p$-algebras are a family of vertex operator algebras
parameterized by $p\in \mathbb Z_{\geq 2}$. They are important examples of
logarithmic 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 called
the $\beta\gamma$ vertex operator algebra.
  We study categories related to the $\mathcal{B}_p$ vertex operator algebras
using their conjectural relation to unrolled restricted quantum groups of
$\mathfrak{sl}_2$. These categories are braided, rigid and non semi-simple
tensor categories. We list their simple and projective objects, their tensor
products and their Hopf links. The latter are successfully compared to modular
data of characters thus confirming a proposed Verlinde formula of David Ridout
and the second author.
