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 unit ary representations of A is a monoidal C*-category with its monoidal unit b eing the GNS representation D-epsilon associated to the counit epsilon. Thi s 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 p ure the dimension function is matrix valued with rows and columns labeled b y the irreducibles contained in D,. This happens precisely when the inclusi ons AL CA and AR CA are not connected. Still, there exists a trace on A whi ch is the Markov trace for both inclusions. We derive two numerical invaria nts for each C*-WHA of trivial hypercenter. These are the common indices I and delta, of the Haar, respectively Markov, conditional expectations of ei ther one of the inclusions A(L/R) subset ofA or (A) over cap (L/R) subset o f(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 ap pear) by showing that I = delta and is always an integer. (C) 2000 Academic Press.