Ambidextrous objects and trace functions for nonsemisimple categories

Geer, Nathan; Kujawa, Jonathan; Patureau-Mirand, Bertrand

doi:10.1090/S0002-9939-2013-11563-7

Ambidextrous objects and trace functions for nonsemisimple categories
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by Nathan Geer, Jonathan Kujawa and Bertrand Patureau-Mirand PDF

Proc. Amer. Math. Soc. 141 (2013), 2963-2978 Request permission

Abstract:

We provide a necessary and sufficient condition for a simple object in a pivotal $\Bbbk$-category to be ambidextrous. In turn, these objects imply the existence of nontrivial trace functions in the category. These functions play an important role in low-dimensional topology as well as in studying the category itself. In particular, we prove they exist for factorizable ribbon Hopf algebras, modular representations of finite groups and their quantum doubles, complex and modular Lie (super)algebras, the $(1,p)$ minimal model in conformal field theory, and quantum groups at a root of unity.

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