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 associated to the counit . This category has isomorphic left dual and right dual objects, which leads, as usual, to the notion of a dimension function. However, if 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 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. Nikshychʹs result (2000, J. Operator Theory, to appear) by showing that I = δ and is always an integer.