Twisting of Quantum Differentials

Abstract

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 : H⊗H → 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.