On braided almost square-free fusion categories with odd dimension

Abstract

Let q be a odd prime number, d be an odd square-free natural number. We prove that a braided fusion category of Frobenius–Perron dimension dqn is solvable. If then we further prove that is tensor equivalent to the category of finite-dimensional representations of a semisimple quasitriangular Hopf algebra which is kG for some abelian group G, or a crossed product for some σ.