On Ribbon Categories for Singlet Vertex Algebras

We construct two non-semisimple braided ribbon tensor categories of modules for each singlet vertex operator algebra M(p), p >= 2. The first category consists of all finite-lengthM(p)-modules with atypical composition factors, while the second is the subcategory of modules that induce to local modules for the triplet vertex operator algebra W(p). We show that every irreducible module has a projective cover in the second of these categories, although not in the first, and we compute all fusion products involving atypical irreducible modules and their projective covers.

Automated summary

This study constructs two non-semisimple braided ribbon tensor categories of modules for each singlet vertex operator algebra M(p), where p>=2. It shows that every irreducible module has a projective cover in the second category and computes all fusion products involving atypical irreducible modules and their projective covers.