## Abstract

Conjugacy classes for groups were generalized in previous works of the authors to semisimple Hopf algebras over algebraically closed fields of characteristic zero. An essential property of the set of conjugacy classes {C_{j}} is that they are irreducible representations of the quantum double D(H) of H. We show that more is true. When (H, R) is quasitriangular, then any irreducible representations of D(H) is a direct summand of a tensor product of the form of {C_{j}} V_{i}, where V_{i} is an irreducible representation of H. The proof follows by applying a partition defined in our 2011 paper on the indices of the irreducible characters of D(H) and identifying the partition with the equivalence classes obtained from Nichols-Richmond equivalence relations on the set of the simple subcoalgebra of D(H)^{∗}. For any block of this partition, there is a unique C_{j} whose character as a D(H)-representation belongs to it. When V is a 1-dimensional representation of H then C_{j} ⊗ V is an irreducible representation of D(H), isomorphic to C_{j} as a vector space endowed with a twisted action of D(H). We apply the above results to H=kG, where G are the groups S_{3} and D_{4}, and decompose all C_{j} ⊗ V_{i} into their irreducible components.

Original language | English |
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Title of host publication | Amitsur Centennial Symposium, 2021 |

Editors | Avinoam Mann, Louis H. Rowen, David J. Saltman, Aner Shalev, Lance W. Small, Uzi Vishne |

Publisher | American Mathematical Society |

Pages | 101-116 |

Number of pages | 16 |

ISBN (Print) | 9781470475550 |

DOIs | |

State | Published - 2024 |

Event | Amitsur Centennial Symposium, 2021 - Jerusalem, Israel Duration: 1 Nov 2021 → 4 Nov 2021 |

### Publication series

Name | Contemporary Mathematics |
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Volume | 800 |

ISSN (Print) | 0271-4132 |

ISSN (Electronic) | 1098-3627 |

### Conference

Conference | Amitsur Centennial Symposium, 2021 |
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Country/Territory | Israel |

City | Jerusalem |

Period | 1/11/21 → 4/11/21 |

### Bibliographical note

Publisher Copyright:© 2024 Miriam Cohen and Sara Westreich.

## Keywords

- Hopf conjugacy classes
- Quantum double; Drinfrld double
- Quasitriangular Hopf algebras