The small quantum group as a quantum double

2009/02/02 by Pavel Etingof, Shlomo Gelaki, Etingof, Pavel +1
Mathematics · Physics and Astronomy · #Advanced Topics in Algebra #Algebraic structures and combinatorial models #Nonlinear Waves and Solitons #math.QA

paper · pdf · doi:10.48550/arxiv.0902.0332

latex, 6 pages; proof of Theorem 4.1 added, a few references added

arxiv created 2009/05/27 · arxiv updated 2009/12/01

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 n² 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.

Citations

• On radically graded finite dimensional quasi-Hopf algebras

Related

• The unrolled quantum group inside Lusztig's quantum group of divided\n powers
• The half-quantum flag variety and representations for small quantum groups
• Small quantum groups associated to Belavin-Drinfeld triples
• The quasi-Hopf analogue of u_q(sl₂)
• Quantum double of a (locally) compact group