Twisting of Quantum Differentials

  • Shahn Majid
  • Robert Oeckl
Conference paper
Part of the Lecture Notes in Physics book series (LNP, volume 543)


We report on recent work (see Majid and Oeckl (1998)). Let H be a Hopf algebra over a field κ. We recall that a unital 2-cocycle χ : HH → κ over H gives rise to a new Hopf algebra H χ (the twist of H) with the same unit, counit and coproduct, but modified product. We show that a bicovariant bimodule V over H can be made a bicovariant bimodule over H χ by equipping it with the same coactions but modified actions. The new (twisted) left action is
$$ a \triangleright _\mathcal{X} b = \mathcal{X}\left( {a_{\left( 1 \right)} \otimes \upsilon _{\left( 1 \right)} } \right)a_{\left( 2 \right)} \triangleright \upsilon _{\left( {\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{2} } \right)} \mathcal{X}^{ - 1} \left( {a_{\left( 3 \right)} \otimes \upsilon _{\left( 3 \right)} } \right), $$
, where the subscripts denote the coproduct or application of the left and the right action.


  1. S. Majid, Hopf Algebras for Physics at the Planck Scale, Class. Quantum Gravity 5 (1988) 1587–1606zbMATHCrossRefADSMathSciNetGoogle Scholar
  2. S. Majid, R. Oeckl, Twisting of quantum differentials and the Planck scale Hopf algebra, DAMTP-1998-118, math/9811054Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2000

Authors and Affiliations

  • Shahn Majid
    • 1
  • Robert Oeckl
    • 1
  1. 1.Department of Applied Mathematics and Theoretical PhysicsUniversity of CambridgeCambridgeUK

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