Fusion Rules Over Dijkgraaf's Deformed Group Double

Diplomarbeit Karolina Vocke 24.12.2010

To answer a question of Prof. Christandl about classifying Sn-anyonmodels in the joint-faculty seminar "Quantum Computing" I aimed to explicitely and systematically describe the fusion rules of the Mod-category over the twisted Drinfel'd double Dw(G) by generalizing Yetter-Drinfel'd modules (action just projective) and characters (now covariantly from G to G).

Karolina Vocke has worked this out as her diploma thesis, which I had the chance to co-supervise, in great detail and for many small examples. We're currently working on a publication. It is now save to say, that the knowledge of conjugacy class fusion rules together with the centralizers projective representation theory easily allows to describe the emerging fusion ring explicitely.

Especially there seems to be an astonishing connection to pointed Hopf algebras: Proper deformability of a groupring (i.e. the simple objects change with an appropriate 3-cocycle) excludes finite dimensional pointed Hopf algebras in all examples we checked so far. The opposite direction is hereby easily disproven by one family of groups of order p3!