Constructing prime modular tensor categories with Hopf algebras

A modular tensor category [Formula: see text] is a non-degenerate ribbon finite tensor category [T. Kerler and V. V. Lyubashenko, Non-Semisimple Topological Quantum Field Theories for 3-Manifolds with Corners, Vol. 1765 (Springer, 2001).]. [Formula: see text] is termed a prime modular tensor category if there exists no non-degenerate braided tensor subcategory other than [Formula: see text] and [Formula: see text] [R. Laugwitz and C. Walton, Constructing non-semisimple modular categories with relative monoidal centers, Int. Math. Res. Not. 2022(20) (2022) 15826–15868.]. This paper aims to construct prime modular tensor categories using methods from Hopf algebra theory. We first present several theorems for constructing ribbon factorizable Hopf algebras to achieve this. Next, we apply these theorems to finite-dimensional pointed Hopf algebras belonging to [Formula: see text], [Formula: see text] introduced by Andruskiewitsch et al., and further establishing conditions under which their associated representation categories are prime modular tensor categories. As concrete applications, we explicitly construct two families of non-semisimple ribbon factorizable Hopf algebras, denoted as [Formula: see text]. By further studying their representation categories, we identify numerous non-semisimple prime modular tensor categories that differ from the representation categories of small quantum groups (i.e. they are not equivalent as tensor categories). Lastly, we construct a family of semisimple ribbon factorizable Hopf algebras, denoted as [Formula: see text], where [Formula: see text] are prime numbers satisfying certain conditions. We then obtain a family of semisimple prime modular tensor categories, which are the Drinfeld doubles of some finite groups, and use them to decompose [Formula: see text] into prime modular tensor categories. We also prove that the Hopf algebras [Formula: see text] cannot be obtained through straightforward means, namely, they are not isomorphic to any tensor products of trivial Hopf algebras (group algebras or their duals) and Drinfeld doubles as Hopf algebras.