2023/08/24 by A. S. N. Chakravarthy, Chakravarthy, Akshaya, Agustina Czenky +3
Mathematics · #Homotopy and Cohomology in Algebraic Topology #Algebraic structures and combinatorial models
paper · pdf · doi:10.48550/arxiv.2308.12546
We contribute to the classification of modular categories C with FPdim(C)≡ 2 \pmod 4. We prove that such categories have group of invertibles of even order, and that they factorize as \mathcal C≅ \widetilde\mathcal C \boxtimes sem, where \widetilde\mathcal C is an odd-dimensional modular category and sem is the rank 2 pointed modular category. This reduces the classification of these categories to the classification of odd-dimensional modular categories. It follows that modular categories \mathcal C with FPdim(C)≡ 2 \pmod 4 of rank up to 46 are pointed. More generally, we prove that if \mathcal C is a weakly integral MTC and p is an odd prime dividing the order of the group of invertibles that has multiplicity one in FPdim(\mathcal C), then we have a factorization \mathcal C ≅ \widetilde\mathcal C \boxtimes Vec\mathbb Zpχ, for \widetilde\mathcal C an MTC with dimension not divisible by p.