Full Centre of an H-Module Algebra

Abstract

We apply the full centre construction, defined in [4], to algebras in and module categories over categories of representations of Hopf algebras.

We obtain a compact formula for the full centre of a module algebra over a Hopf algebra.

Key Words:

• Full centre
• Hopf algebra
• Monoidal category
• Yetter–Drinfeld module

2000 Mathematics Subject Classification:

• Primary 18D10
• 16T05

ACKNOWLEDGMENTS

The work on the note began during the author's visit to São Paulo, where the author was accommodated by the Institute of Mathematics and Statistics of the University of São Paulo, was continued in 2009, while the author was visiting Macquarie University, and finished at Max Planck Institut Für Mathematik (Bonn) in winter 2010. The author would like to thank these institutions for hospitality and excellent working conditions. The author would like to thank FAPESP (grant no. 2008/10526-1), Safety Net Grant of Macquarie University, and Max Planck Gesellschaft, whose financial support made the visits to Bonn, São Paulo, and Sydney possible. The idea to write the note came to the author during the “Colloquium on Hopf algebras, Quantum groups, and Tensor categories” (La Falda, Argentina, September 2009). The author thanks Nicolás Andruskiewitsch for the invitation. The author also would like to thank Jürgen Fuchs, Vyacheslav Futorny, Sonia Natale, Ingo Runkel, and York Sommerhäuser for inspiring conversations. The author thanks Gigel Militaru and York Sommerhäuser for pointing out the references [1 9-11]. Special thanks are to J. Fuchs for spotting a number of misprints in the preliminary version of the article.

Notes

Communicated by V. A. Artamonov.