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 Dε associated to the counit ε. This category has isomorphic left dual and right dual objects, which leads, as usual, to the notion of a dimension function. However, if ε is not pure the dimension function is matrix valued with rows and columns labeled by the irreducibles contained in Dε. This happens precisely when the inclusions AL⊂A and AR⊂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 δ, of the Haar, respectively Markov, conditional expectations of either one of the inclusions AL/R⊂A or ÁL/R⊂Á. In generic cases I>δ. In the special case of weak Kac algebras we reproduce D. Nikshych's result (2000, J. Operator Theory, to appear) by showing that I=δ and is always an integer.