AN INTRODUCTION TO NONCOMMUTATIVE DIFFERENTIAL GEOMETRY ON QUANTUM GROUPS

We give a pedagogical introduction to the differential calculus on quantum groups by stressing at all stages the connection with the classical case (q --> 1 limit). The Lie derivative and the contraction operator on forms and tensor fields are found. A new, explicit form of the Cartan-Maurer equations is presented. The example of a bicovariant differential calculus on the quantum group GL(q)(2) is given in detail. The softening of a quantum group is considered, and we introduce q curvatures satisfying q Bianchi identities, a basic ingredient for the construction of q gravity and q gauge theories.