Communications in Mathematical Physics

, Volume 199, Issue 3, pp 547–589

Doubles of Quasi-Quantum Groups

  • Frank Hausser
  • Florian Nill

DOI: 10.1007/s002200050512

Cite this article as:
Hausser, F. & Nill, F. Comm Math Phys (1999) 199: 547. doi:10.1007/s002200050512

Abstract:

In [Dr1] Drinfeld showed that any finite dimensional Hopf algebra \(\) extends to a quasitriangular Hopf algebra \(\), the quantum double of \(\). Based on the construction of a so-called diagonal crossed product developed by the authors in [HN], we generalize this result to the case of quasi-Hopf algebras \(\). As for ordinary Hopf algebras, as a vector space the “quasi-quantum double”\(\) is isomorphic to \(\), where \(\) denotes the dual of \(\). We give explicit formulas for the product, the coproduct, the R-matrix and the antipode on \(\) and prove that they fulfill Drinfeld's axioms of a quasitriangular quasi-Hopf algebra. In particular \(\) becomes an associative algebra containing \(\) as a quasi-Hopf subalgebra. On the other hand, \(\) is not a subalgebra of \(\) unless the coproduct on \(\) is strictly coassociative. It is shown that the category \(\) of finite dimensional representations of \(\) coincides with what has been called the double category of \(\)-modules by S. Majid [M2]. Thus our construction gives a concrete realization of Majid's abstract definition of quasi-quantum doubles in terms of a Tannaka–Krein-like reconstruction procedure. The whole construction is shown to generalize to weak quasi-Hopf algebras with \(\) now being linearly isomorphic to a subspace of \(\).

Copyright information

© Springer-Verlag Berlin Heidelberg 1999

Authors and Affiliations

  • Frank Hausser
    • 1
  • Florian Nill
    • 1
  1. 1.Freie Universität Berlin, Institut für Theoretische Physik, Arnimallee 14, D-14195 Berlin, Germany.¶E-mail: hausser@physik.fu-berlin.de; nill@physik.fu-berlin.deDE