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Preprints, Working Papers, ... Year : 2018

Modularization of small quantum groups

Azat M. Gainutdinov
Simon Lentner
  • Function : Author
Tobias Ohrmann
  • Function : Author


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].

Dates and versions

hal-01880816 , version 1 (25-09-2018)



Azat M. Gainutdinov, Simon Lentner, Tobias Ohrmann. Modularization of small quantum groups. 2018. ⟨hal-01880816⟩
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