Weak Hopf Algebras: II. Representation Theory, Dimensions, and the Markov Trace
✍ Scribed by Gabriella Böhm; Kornél Szlachányi
- Publisher
- Elsevier Science
- Year
- 2000
- Tongue
- English
- Weight
- 371 KB
- Volume
- 233
- Category
- Article
- ISSN
- 0021-8693
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✦ Synopsis
If A is a weak C U -Hopf algebra then the category of finite-dimensional unitary representations of A is a monoidal C U -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 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 U -WHA of trivial hypercenter. These are the common indices I and ␦ , of the Haar, respectively Markov, conditional expecta-L r R ˆL r R tions of either one of the inclusions A ; A or A ; A. In generic cases I ) ␦. In the special case of weak Kac algebras we reproduce D. Nikshych's result Ž . 2000, J. Operator Theory, to appear by showing that I s ␦ and is always an integer.