TABLE OF CONTENTS Quantum groups are particular Hopf algebras (see Section 10.2) and, consequently, particular non-commutative algebras provided with co-operations. There are various definitions of quantum groups. The most of them treats a quantum group as a certain Hopf algebra generated by elements assembled into a matrix whose entries are not numbers. Hopf algebras and quantum groups make a contribution to many quantum models [97; 249; 292; 293]. However, the development of differential calculus and differential geometry over Hopf algebras has met problems. Here, we are concerned with the two of them. These are differential calculus over Hopf algebras (Section 9.2) and quantum principal bundles (Sections 9.3). In the case of Lie groups, there are two equivalent definitions of a smooth principal bundle, which is both a set of trivial bundles glued together by means of transition functions and a bundle provided with the canonical action of a structure group on the right. In the case of quantum groups, these two notions of a principal bundle are not matched in general.
Let us compile different definitions of quantum groups [97; 98; 249; 293].
In a general setting, any non-cocommutative Hopf algebra can be treated as a quantum group. For the sake of convenience, we call them general quantum groups. There are several methods of constructing general quantum groups. For instance, the following one is utilized in order to obtain quantum deformations of simple complex Lie algebras [97]. Let ( 
, m, ?, ?, S) be...
