The small quantum group as a quantum double

We prove that the quantum double of the quasi-Hopf algebra Aq(g) of dimension ndimg attached in [P. Etingof, S. Gelaki, On radically graded finite-dimensional quasi-Hopf algebras, Mosc. Math. J. 5 (2) (2005) 371–378] to a simple complex Lie algebra g and a primitive root of unity q of order n2 is equivalent to Lusztig's small quantum group uq(g) (under some conditions on n). We also give a conceptual construction of Aq(g) using the notion of de-equivariantization of tensor categories.