An introduction to non-commutative differential geometry on quantum groups

TL;DR

This paper provides a comprehensive introduction to non-commutative differential geometry on quantum groups, connecting classical and quantum cases, and introduces new formulations of Cartan--Maurer equations and q-curvatures relevant for quantum gravity and gauge theories.

Contribution

It offers a pedagogical overview of differential calculus on quantum groups, including explicit formulas and the concept of q-curvatures, advancing the understanding of quantum geometric structures.

Findings

Derived explicit Cartan--Maurer equations for quantum groups

Introduced q-curvatures satisfying q-Bianchi identities

Detailed example of bicovariant calculus on $GL_q(2)$

Abstract

We give a pedagogical introduction to the differential calculus on quantum groups by stressing at all stages the connection with the classical case ($q \rightarrow 1$ 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.