Frobenius-Perron Dimensions of Integral \(\mathbb {Z}_{+}\)-rings and Applications

Abstract

We introduce the notion of the Frobenius-Perron dimension of an integral \(\mathbb {Z}_{+}\)-ring and give some applications of this notion to classification of finite dimensional quasi-Hopf algebras with a unique nontrivial simple module, and of quasi-Hopf and Hopf algebras of prime dimension p.

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Acknowledgements

The author is grateful to C. Negron for useful discussions and to V. Ostrik, S.-H. Ng and X. Wang for corrections and comments on the draft of this paper. The work of the author was partially supported by the NSF grant DMS-1502244.

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Correspondence to Pavel Etingof.

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To Nicolás Andruskiewitsch on his 60th birthday with admiration

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Presented by: Alistair Savage

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Etingof, P. Frobenius-Perron Dimensions of Integral \(\mathbb {Z}_{+}\)-rings and Applications. Algebr Represent Theor 23, 2059–2078 (2020). https://doi.org/10.1007/s10468-019-09924-1

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Keywords

  • Hopf algebra
  • quasi-Hopf algebra
  • Frobenius-Perron dimension

Mathematics Subject Classification (2010)

  • 16T05
  • 18D10