An introduction to non-commutative differential geometry on quantum groups

Abstract

We give a pedagogical introduction to the differential calculus on quantum groups by stressing at all stages the connection with the classical case ($q\to 1$ 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 $G{L}_q(2)$ 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.

45 pages

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