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Covariantization of quantized calculi over quantum groups

Abstract. We introduce a method for construction of a covariant differential calculus over a Hopf algebra A from a quantized calculus da = [D, a], a ∈ A, where D is a candidate for a Dirac operator for A. We recover the method of construction of a bicovariant differential calculus given by T.Brzeziński and S.Majid created from a central element of the dual Hopf algebra A. We apply this method to the Dirac operator for the quantum SL(2) given by S.Majid. We find that the differential calculus obtained by our method is the standard bicovariant 4D-calculus. We also apply this method to the Dirac operator for the quantum SL(2) given by P.N.Bibikov and P.P.Kulish and show that the resulted differential calculus is 8-dimensional.