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

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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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References

  1. Chebolu, S., Lockridge, K.: Fields with indecomposable multiplicative groups. Expo. Math. 34, 237–242 (2016). arXiv:1407.3481

    Article  MathSciNet  Google Scholar 

  2. Creutzig, T., Gainutdinov, A.M., Runkel, I.: A quasi-Hopf algebra for the triplet vertex operator algebra. arXiv:1712.07260

  3. Etingof, P., Gelaki, S., Nikshych, D., Ostrik, V.: Tensor Categories. AMS, Providence (2015)

    Book  Google Scholar 

  4. Etingof, P., Gelaki, S.: Quasisymmetric and unipotent tensor categories. Math. Res. Lett. 15(5), 857–866 (2008)

    Article  MathSciNet  Google Scholar 

  5. Etingof, P., Gelaki, S.: Finite dimensional quasi-Hopf algebras with radical of codimension 2. MRL 11(5), 685–696 (2004)

    MathSciNet  MATH  Google Scholar 

  6. Etingof, P., Gelaki, S.: On finite-dimensional semisimple and cosemisimple Hopf algebras in positive characteristic, IMRN, 1998, Issue 16, pp. 851–864

  7. Dolfi, S., Navarro, G.: Finite groups with only one nonlinear irreducible representation. Comm. Algebra 40(11), 4324–4329 (2012)

    Article  MathSciNet  Google Scholar 

  8. Gainutdinov, A.M., Runkel, I.: Projective objects and the modified trace in factorisable finite tensor categories. arXiv:1703.00150

  9. Negron, C.: Log-modular quantum groups at an even root of unity and quantum Frobenius I. arXiv:1812.02277

  10. Nichols, W.: Bialgebras of type one. Comm. Algebra 6(15), 1521–1552 (1978)

    Article  MathSciNet  Google Scholar 

  11. Ng, R., Wang, X.: Hopf algebras of prime dimension in positive characteristic, arXiv:1810.00476

  12. Nguyen, V.C., Wang, L., Wang, X.: Classification of connected Hopf algebras of dimension p3, I, arXiv:1309.0286

  13. Wang, X.: Connected Hopf algebras of dimension p2. arXiv:1208.2280 (2280)

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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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  1. Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA, 02139, USA

    Pavel Etingof

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  1. Pavel Etingof
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Correspondence to Pavel Etingof.

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Presented by: Alistair Savage

To Nicolás Andruskiewitsch on his 60th birthday with admiration

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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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  • DOI: https://doi.org/10.1007/s10468-019-09924-1

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