Cartan Calculus: Differential Geometry for Quantum Groups

The topic of this lecture is differential geometry on quantum groups. Several lecturers at this conference have talked about this subject. For a review and a fairly extensive list of references I would like to point the reader to the contributions of B. Jurco and S. L. Woronowicz in this proceedings. Here we shall propose a new rigid framework for the so-called Cartan calculus of Lie derivatives, inner derivations, functions, and forms. The construction employs a semi-direct product of two graded Hopf algebras, the respective super-extensions of the deformed universal enveloping algebra and the algebra of functions on a quantum group. All additional relations in the Cartan calculus follow as consistency conditions. The approach is not based on the Leibniz rule for the exterior derivative and might hence also be of interest in the recent work on its deformations. However, given a d that satisfies the Leibniz rule, the Cartan identity (77) follows as a theorem. Rigorous proofs for many statements in this presentation can be found in some form or other in [1]. For a nice review see e.g. [2]. For Quantum Groups and (quasitriangular) Hopf algebras one could consult [3, 4]. In the next section we would like to motivate the semi-direct product construction by some geometrical considerations.