  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].
