The Relations of Inner and Outer Differential Calculi on Quantum Groups

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 ${\cal A}$, are related by homomorphisms, which map the elements of one class on elements of the other class.