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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
Article

Abstract

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.

Keywords

Neural Network Statistical Physic Complex System Nonlinear Dynamics Differential Geometry 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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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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