Weak Hopf Algebras II: Representation theory, dimensions and the Markov trace

If A is a weak C^*-Hopf algebra then the category of finite dimensional unitary representations of A is a monoidal C^*-category with monoidal unit being the GNS representation D_eps associated to the counit \eps. This category has isomorphic left dual and right dual objects which leads, as usual, to the notion of dimension function. However, if \eps is not pure the dimension function is matrix valued with rows and columns labelled by the irreducibles contained in D_eps. This happens precisely when the inclusions A^L < A and A^R < A are not connected. Still there exists a trace on A which is the Markov trace for both inclusions. We derive two numerical invariants for each C^*-WHA of trivial hypercenter. These are the common indices I and \delta, of the Haar, respectively Markov conditional expectations of either one of the inclusions A^{L/R} < A and Adual^{L/R} < Adual. In generic cases I > \delta. In the special case of weak Kac algebras we show that I=\delta is an integer.