The small quantum group as a quantum double

TL;DR

This paper demonstrates that the quantum double of a specific quasi-Hopf algebra related to a Lie algebra is equivalent to Lusztig's small quantum group, providing a new conceptual construction via tensor category de-equivariantization.

Contribution

It establishes an equivalence between the quantum double of A_q(g) and the small quantum group u_q(g), and introduces a new categorical construction method.

Findings

Quantum double of A_q(g) is equivalent to u_q(g).

Provides a conceptual categorical construction of A_q(g).

Conditions on n for the equivalence are specified.

Abstract

We prove that the quantum double of the quasi-Hopf algebra A_q(g) of dimension n^{dim g} attached in arXiv:math/0403096 to a simple complex Lie algebra g and a primitive root of unity q of order n^2 is equivalent to Lusztig's small quantum group u_q(g) (under some conditions on n). We also give a conceptual construction of A_q(g) using the notion of de-equivariantization of tensor categories.