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Ambidextrous objects and trace functions for nonsemisimple categories


Authors: Nathan Geer, Jonathan Kujawa and Bertrand Patureau-Mirand
Journal: Proc. Amer. Math. Soc. 141 (2013), 2963-2978
MSC (2010): Primary 18D10; Secondary 17B99, 16T05, 20C20, 57M99
DOI: https://doi.org/10.1090/S0002-9939-2013-11563-7
Published electronically: May 10, 2013
MathSciNet review: 3068949
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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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Additional Information

Nathan Geer
Affiliation: Department of Mathematics and Statistics, Utah State University, Logan, Utah 84322
Email: nathan.geer@usu.edu

Jonathan Kujawa
Affiliation: Department of Mathematics, University of Oklahoma, Norman, Oklahoma 73019
Email: kujawa@math.ou.edu

Bertrand Patureau-Mirand
Affiliation: LMAM, Université de Bretagne-Sud, Université Européenne de Bretagne, BP 573, 56017 Vannes, France
Email: bertrand.patureau@univ-ubs.fr

DOI: https://doi.org/10.1090/S0002-9939-2013-11563-7
Received by editor(s): June 22, 2011
Received by editor(s) in revised form: November 15, 2011
Published electronically: May 10, 2013
Additional Notes: Research of the first author was partially supported by NSF grants DMS-0968279 and DMS-1007197.
Research of the second author was partially supported by NSF grant DMS-0734226 and NSA grant H98230-11-1-0127.
Communicated by: Kailash E. Misra
Article copyright: © Copyright 2013 American Mathematical Society