DERIVATIONS WITH QUANTUM GROUP ACTION

ABSTRACT

The derivations of a left coideal subalgebra ℬ of a Hopf algebra 𝒜 which are compatible with the comultiplication of 𝒜 (that is, the covariant first order differential calculi, as defined by Woronowicz, on a quantum homogeneous space) are related to certain right ideals of ℬ. The correspondence is one-to-one if 𝒜 is faithfully flat as a right ℬ-module. This generalizes the result for ℬ=𝒜 due to Woronowicz. A definition for the dimension of a first order differential calculus at a classical point is given. For the quantum 2-sphere of Podleś under the assumptions and for all n=0, 1,…, three 2-dimensional covariant first order differential calculi exist if c=0, one exists if and none else. This extends a result of Podleś.

ACKNOWLEDGMENTS

I am grateful to Prof. K. Schmüdgen for showing his interest in my work and to S. Kolb for detailed discussions. This work was supported by the Deutsche Forschungsgemeinschaft within the scope of the postgraduate scholarship programme “Graduiertenkolleg Quantenfeldtheorie” at the University of Leipzig.