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 its monoidal unit being the GNS representation D-epsilon associated to the counit epsilon. This category has isomorphic left dual and right dual objects, which leads, as usual, to the notion of a dimension function. However, if epsilon is not pure the dimension function is matrix valued with rows and columns labeled by the irreducibles contained in D,. This happens precisely when the inclusions AL CA and AR CA 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) subset ofA or (A) over cap (L/R) subset of(A) over cap. In generic cases I > delta. In the special case of weak Kac algebras we reproduce D. Nikshych's result (2000, J. Operator Theory, to appear) by showing that I = delta and is always an integer. (C) 2000 Academic Press.