Modular tensor categories arising from central extensions and related applications

Abstract

A modular tensor category is a non-degenerate ribbon finite tensor category and a ribbon factorizable Hopf algebra is a Hopf algebra whose finite-dimensional representations form a modular tensor category. In this paper, we provide a method of constructing ribbon factorizable Hopf algebras using central extensions. We then apply this method to n-rank Taft algebras, which are considered finite-dimensional quantum groups associated with abelian Lie algebras (see Section 2 for the definition), and obtain a family of non-semisimple ribbon factorizable Hopf algebras $E_q$, thus producing non-semisimple modular tensor categories using their representation categories. And we provide a prime decomposition of $\mathrm{Rep}\left(E_q\right)$ (the representation category of $E_q$). By further studying the simplicity of $E_q$ (whether it is a simple Hopf algebra or not), we conclude that

• (1)

  there exists a twist J of $u_q\left(s{l}_2^{\oplus 3}\right)$ such that $u_q{\left(s{l}_2^{\oplus 3}\right)}^J$ is a simple Hopf algebra,

• (2)

  there is no relation between the simplicity of a Hopf algebra H and the primality of $\mathrm{Rep}\left(H\right)$,

• (3)

  there are many ribbon factorizable Hopf algebras that are distinct from some known ones, i.e., not isomorphic to any tensor products of trivial Hopf algebras (group algebras or their dual), Drinfeld doubles, and small quantum groups.