Communications in Mathematical Physics

, Volume 149, Issue 3, pp 513–548 | Cite as

Action of truncated quantum groups on quasi-quantum planes and a quasi-associative differential geometry and calculus

  • Gerhard Mack
  • Volker Schomerus


Ifq is apth root of unity there exists a quasi-coassociative truncated quantum group algebra whose indecomposable representations are the physical representations ofU q (sl2), whose coproduct yields the truncated tensor product of physical representations ofU q (sl2), and whoseR-matrix satisfies quasi-Yang Baxter equations. These truncated quantum group algebras are examples of weak quasitriangular quasi-Hopf algebras (“quasi-quantum group algebras”)
. We describe a space
of “functions on the quasi quantum plane,” i.e. of polynomials in noncommuting complex coordinate functionsz a , on which multiplication operatorsZ a and the elements of
can act, so thatz a will transform according to some representation τf of
can be made into a quasi-associative graded algebra
on which elements of
act as generalized derivations. In the special case of the truncatedU q (sl2) algebra we show that the subspaces
of monomials inz a ofnth degree vanish forn≥p−1, and that
carries the 2J+ 1 dimensional irreducible representation of
ifn=2J, J=0,1/2, ..., 1/2(p−2). Assuming that the representation τf of the quasi-quantum group algebra gives rise to anR-matrix with two eigenvalues, we develop a quasi-associative differential calculus on
. This implies construction of an exterior differentiation, a graded algebra
of forms and partial derivatives. A quasi-associative generalization of noncommutative differential geometry is introduced by defining a covariant exterior differentiation of forms. It is covariant under
gauge transformations.


Neural Network Statistical Physic Complex System Nonlinear Dynamics Differential Geometry 
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  1. 1.
    Manin, Yu.I.: Quantum groups and Non-commutative geometry. Proc. Int. Congr. Math., Berkeley1, 798 (1986)Google Scholar
  2. 2.
    Wess, J., Zumino, B.: Covariant differential calculus on the quantum hyperplane. Preprint CERN TH 5697/90, April 1990Google Scholar
  3. 3.
    Woronowicz, S.L.: Differential calculus on compact matrix pseudogroups (quantum groups). Commun. Math. Phys.122, 125 (1989)Google Scholar
  4. 4.
    Drinfel'd, V.G.: Quantum groups. Proc. ICM 798 (1987)Google Scholar
  5. 5.
    Drinfel'd, V.G.: Quasi-Hopf algebras and Knizhnik Zamolodchikov equations. In: Problems of modern quantum field theory, Proceedings Alushta 1989, Research reports in Physics. Berlin, Heidelberg, New York: Springer 1989Google Scholar
  6. 6.
    Mack, G., Schomerus, V.: In preparationGoogle Scholar
  7. 7.
    Mack, G., Schomerus, V.: Quasi-Hopf Quantum Symmetry in Quantum Theory. Nucl. Phys. B370, 185 (1992)Google Scholar
  8. 8.
    Mack, G., Schomerus, V.: Quasi-quantum group symmetry and local braid relations in the conformal Ising model. Phys. Lett. B267, 207 (1991)Google Scholar
  9. 9.
    Sudbury, A.: Non-commuting coordinates and differential operators. In: Quantum groups, Curtright, T., et al. (eds.) Singapore: World Scientific 1991Google Scholar
  10. 10.
    Reshetikhin, N.Yu., Takhtajan, L.A., Faddeev, L.D.: Algebra Analysis1, 178 (1989)Google Scholar
  11. 11.
    Connes, A.: Noncommutative differential geometry. Publ. Math. I.H.E.S.62 (1985)Google Scholar

Copyright information

© Springer-Verlag 1992

Authors and Affiliations

  • Gerhard Mack
    • 1
  • Volker Schomerus
    • 1
  1. 1.II. Institut für Theoretische PhysikUniversität HamburgHamburg 50FRG

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