Modularization of small quantum groups

TL;DR

This paper constructs a broad class of ribbon quasi-Hopf algebras related to small quantum groups to produce non-semisimple modular tensor categories at even roots of unity, extending the understanding of quantum group representations.

Contribution

It introduces explicit constructions of quasi-Hopf algebras from small quantum groups, providing new modularizations and explicit generators, and explores their applications to vertex operator algebras and braided categories.

Findings

Constructed a family of ribbon quasi-Hopf algebras with factorizable R-matrices.

Established modular tensor categories from these quasi-Hopf algebras at even roots of unity.

Connected the algebraic structures to vertex operator algebras and conjectured equivalences of categories.

Abstract

We construct a large family of ribbon quasi-Hopf algebras related to small quantum groups, with a factorizable R-matrix. Our main purpose is to obtain non-semisimple modular tensor categories for quantum groups at even roots of unity, where typically the initial representation category is not even braided. Our quasi-Hopf algebras are built from modules over the twisted Drinfeld double via a universal construction, but we also work out explicit generators and relations, and we prove that these algebras are modularizations of the quantum group extensions with R-matrices listed in [LO17]. As an application, we find one distinguished factorizable quasi-Hopf algebra for any finite root system and any root of unity of even order (resp. divisible by 4 or 6, depending on the root length). Under the same divisibility condition on a rescaled root lattice, a corresponding lattice Vertex-Operator Algebra contains a VOA W defined as the kernel of screening operators. We then conjecture that W representation categories are braided equivalent to the representation categories of the distinguished factorizable quasi-Hopf algebras. For A_1 root system, our construction specializes to the quasi-Hopf algebras in [GR17b, CGR17], where the answer is affirmative, similiary for B_n at fourth root of unity in [FGR17b, FL17].