### Abstract

Kashina, Montgomery and Ng introduced the n-th indicator ν_{n}(H) of a finite-dimensional Hopf algebra H and showed that the indicators have some interesting properties such as the gauge invariance. The aim of this paper is to investigate the properties of ν_{n}’s. In particular, we obtain the cyclotomic integrality of ν_{n} and a formula for ν_{n} of the Drinfeld double. Our results are applied to the finite-dimensional pointed Hopf algebra u(D, λ, µ) introduced by Andruskiewitsch and Schneider. As an application, we obtain the second indicator of u_{q}(sl_{2}) and show that if p and q are roots of unity of the same order, then u_{p}(sl_{2}) and u_{q}(sl_{2}) are gauge equivalent if and only if q = p, where p and q are roots of unity of the same odd order.

Original language | English |
---|---|

Pages (from-to) | 155-201 |

Number of pages | 47 |

Journal | Israel Journal of Mathematics |

Volume | 207 |

Issue number | 1 |

DOIs | |

Publication status | Published - 2015 Feb 17 |

Externally published | Yes |

### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

*Israel Journal of Mathematics*,

*207*(1), 155-201. https://doi.org/10.1007/s11856-015-1156-x

**On indicators of Hopf algebras.** / Shimizu, Kenichi.

Research output: Contribution to journal › Article

*Israel Journal of Mathematics*, vol. 207, no. 1, pp. 155-201. https://doi.org/10.1007/s11856-015-1156-x

}

TY - JOUR

T1 - On indicators of Hopf algebras

AU - Shimizu, Kenichi

PY - 2015/2/17

Y1 - 2015/2/17

N2 - Kashina, Montgomery and Ng introduced the n-th indicator νn(H) of a finite-dimensional Hopf algebra H and showed that the indicators have some interesting properties such as the gauge invariance. The aim of this paper is to investigate the properties of νn’s. In particular, we obtain the cyclotomic integrality of νn and a formula for νn of the Drinfeld double. Our results are applied to the finite-dimensional pointed Hopf algebra u(D, λ, µ) introduced by Andruskiewitsch and Schneider. As an application, we obtain the second indicator of uq(sl2) and show that if p and q are roots of unity of the same order, then up(sl2) and uq(sl2) are gauge equivalent if and only if q = p, where p and q are roots of unity of the same odd order.

AB - Kashina, Montgomery and Ng introduced the n-th indicator νn(H) of a finite-dimensional Hopf algebra H and showed that the indicators have some interesting properties such as the gauge invariance. The aim of this paper is to investigate the properties of νn’s. In particular, we obtain the cyclotomic integrality of νn and a formula for νn of the Drinfeld double. Our results are applied to the finite-dimensional pointed Hopf algebra u(D, λ, µ) introduced by Andruskiewitsch and Schneider. As an application, we obtain the second indicator of uq(sl2) and show that if p and q are roots of unity of the same order, then up(sl2) and uq(sl2) are gauge equivalent if and only if q = p, where p and q are roots of unity of the same odd order.

UR - http://www.scopus.com/inward/record.url?scp=84931577991&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84931577991&partnerID=8YFLogxK

U2 - 10.1007/s11856-015-1156-x

DO - 10.1007/s11856-015-1156-x

M3 - Article

AN - SCOPUS:84931577991

VL - 207

SP - 155

EP - 201

JO - Israel Journal of Mathematics

JF - Israel Journal of Mathematics

SN - 0021-2172

IS - 1

ER -