The B p -algebras are a family of vertex operator algebras parameterized by p ∈ Z ≥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 B 2 -algebra, are the well-known symplectic bosons also often called the βγ vertex operator algebra.We study categories related to the B p vertex operator algebras using their conjectural relation to unrolled restricted quantum groups of sl 2 . These categories are braided, rigid and non semisimple tensor categories. We list their simple and projective objects, their tensor products and their Hopf links. The latter are succesfully compared to modular data of characters thus confirming a proposed Verlinde formula of David Ridout and the second author.

The parafermionic cosets C k = Com(H,L k (sl 2 )) are studied for negative admissible levels k, as are certain infinite-order simple current extensions B k of C k . Under the assumption that the tensor theory considerations of Huang, Lepowsky and Zhang apply to C k , irreducible C k -and B k -modules are obtained from those of L k (sl 2 ). Assuming the validity of a certain Verlinde-type formula likewise gives the Grothendieck fusion rules of these irreducible modules. Notably, there are only finitely many irreducible B k -modules. The irreducible C k -and B k -characters are computed and the latter are shown, when supplemented by pseudotraces, to carry a finite-dimensional representation of the modular group. The natural conjecture then is that the B k are C 2 -cofinite vertex operator algebras.[math.QA].

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We construct a direct sum completion C ⊕ of a given braided monoidal category C which allows for the rigorous treatment of infinite order simple current extensions of vertex operator algebras as seen in [CKL]. As an example, we construct the vertex operator algebra V L associated to an even lattice L as an infinite order simple current extension of the Heisenberg VOA and recover the structure of its module category through categorical considerations.

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