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 S(q,c) of Podles under the assumptions "q is not a root of unity" and "c is not equal to -q^(2n)/(q^(2n)+1)^2" for all n=0,1,..., three 2-dimensional covariant first order differential calculi exist if c=0, one exists if c=-q/(q+1)^2 or c=q/(-q+1)^2 and none else. This extends a result of Podles.