Classification of Differential Calculi on U_q(b+), Classical Limits, and Duality

We give a complete classification of bicovariant first order differential calculi on the quantum enveloping algebra U_q(b+) which we view as the quantum function algebra C_q(B+). Here, b+ is the Borel subalgebra of sl_2. We do the same in the classical limit q->1 and obtain a one-to-one correspondence in the finite dimensional case. It turns out that the classification is essentially given by finite subsets of the positive integers. We proceed to investigate the classical limit from the dual point of view, i.e. with ``function algebra'' U(b+) and ``enveloping algebra'' C(B+). In this case there are many more differential calculi than coming from the q-deformed setting. As an application, we give the natural intrinsic 4-dimensional calculus of kappa-Minkowski space and the associated formal integral.