## Abstract

We offer a comprehensive discussion on Verlinde-type formulas for Hopf algebras H over an algebraically closed field of characteristic 0. Some of the results are new and some are known, but are reproved from the point of view of symmetric algebras and the associated Higman (trace) map. We give an explicit form for the central Casimir element of C(H), which is also known to be χad, the character of the adjoint map on H. We then discuss the following variations of the Verlinde formula: (i) Fusion rules for irreducible characters of semisimple Hopf algebras whose character algebras C(H) are commutative. (ii) Structure constants for what we call here conjugacy sums associated to conjugacy classes for these Hopf algebras. (iii) Equality up to rational scalar multiples between the fusion rules of irreducible characters and the structure constants for semisimple factorizable Hopf algebras. (iv) Projective fusion rules for the multiplication of irreducible and indecomposable projective characters for non-semisimple factorizable Hopf algebras.

Original language | English |
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Title of host publication | Ring and Module Theory |

Editors | Toma Albu, Gary F. Birkenmeier, Ali Erdoǧan, Adnan Tercan |

Publisher | Springer International Publishing |

Pages | 91-114 |

Number of pages | 24 |

ISBN (Print) | 9783034600064 |

DOIs | |

State | Published - 2010 |

Event | International Conference on Ring and Module Theory, 2008 - Ankara, Turkey Duration: 18 Aug 2008 → 22 Aug 2008 |

### Publication series

Name | Trends in Mathematics |
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Volume | 50 |

ISSN (Print) | 2297-0215 |

ISSN (Electronic) | 2297-024X |

### Conference

Conference | International Conference on Ring and Module Theory, 2008 |
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Country/Territory | Turkey |

City | Ankara |

Period | 18/08/08 → 22/08/08 |

### Bibliographical note

Publisher Copyright:© 2010 Springer Basel AG.

## Keywords

- Characters
- Conjugacy sums and classes
- Factorizable ribbon Hopf algebras
- Fusion rules
- Hopf algebras
- Projective center
- Symmetric algebras
- Verlinde formula