The Relations of Inner and Outer Differential Calculi on Quantum Groups

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The differential caluli (Gamma,d) on quantum groups are classified due to the property of the generating element X of its differential d. There are, on the one hand differential caluli which contain this element X in the basis of one- forms that span Gamma, called Inner Differential Calculi. On the other hand, one has the differential caluli which do not contain the generating element X of its differential d, thus they are called Outer Differential Calculi. We show that this two classes of differential caluli, for a given quantum group A, are related by homomorphisms, which map the elements of one class on elements of the other class.