Communications in Mathematical Physics

, Volume 122, Issue 1, pp 125–170 | Cite as

Differential calculus on compact matrix pseudogroups (quantum groups)

  • S. L. Woronowicz


The paper deals with non-commutative differential geometry. The general theory of differential calculus on quantum groups is developed. Bicovariant bimodules as objects analogous to tensor bundles over Lie groups are studied. Tensor algebra and external algebra constructions are described. It is shown that any bicovariant first order differential calculus admits a natural lifting to the external algebra, so the external derivative of higher order differential forms is well defined and obeys the usual properties. The proper form of the Cartan Maurer formula is found. The vector space dual to the space of left-invariant differential forms is endowed with a bilinear operation playing the role of the Lie bracket (commutator). Generalized antisymmetry relation and Jacobi identity are proved.


Quantum Group Differential Form Jacobi Identity Differential Calculus Proper Form 
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Copyright information

© Springer-Verlag 1989

Authors and Affiliations

  • S. L. Woronowicz
    • 1
  1. 1.Department of Mathematical Methods in Physics, Faculty of PhysicsUniversity of WarsawWarszawaPoland

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