### Abstract

By weakening the counit and antipode axioms of a C^{*}-Hopf algebra and allowing for the coassociative coproduct to be nonunital, we obtain a quantum group, that we call a weak ^{C*}-Hopf algebra, which is sufficiently general to describe the symmetries of essentially arbitrary fusion rules. This amounts to generalizing the Baaj-Skandalis multiplicative unitaries to multipicative partial isometrics. Every finite-dimensional weak C^{*}-Hopf algebra has a dual which is again a weak C^{*}-Hopf algebra. An explicit example is presented with Lee-Yang fusion rules. We briefly discuss applications to amalgamated crossed products, doubles, and quantum chains.

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

Pages (from-to) | 437-456 |

Number of pages | 20 |

Journal | Letters in Mathematical Physics |

Volume | 38 |

Issue number | 4 |

Publication status | Published - 1996 |

### Fingerprint

### Keywords

- Amalgamated crossed products
- Multiplicative isometrics
- Weak Hopf algebras

### ASJC Scopus subject areas

- Mathematical Physics
- Physics and Astronomy(all)
- Statistical and Nonlinear Physics

### Cite this

*Letters in Mathematical Physics*,

*38*(4), 437-456.

**A Coassociative C-Quantum Group with Nonintegral Dimensions.** / Böhm, G.; Szlachónyi, Korníl.

Research output: Contribution to journal › Article

*Letters in Mathematical Physics*, vol. 38, no. 4, pp. 437-456.

}

TY - JOUR

T1 - A Coassociative C-Quantum Group with Nonintegral Dimensions

AU - Böhm, G.

AU - Szlachónyi, Korníl

PY - 1996

Y1 - 1996

N2 - By weakening the counit and antipode axioms of a C*-Hopf algebra and allowing for the coassociative coproduct to be nonunital, we obtain a quantum group, that we call a weak C*-Hopf algebra, which is sufficiently general to describe the symmetries of essentially arbitrary fusion rules. This amounts to generalizing the Baaj-Skandalis multiplicative unitaries to multipicative partial isometrics. Every finite-dimensional weak C*-Hopf algebra has a dual which is again a weak C*-Hopf algebra. An explicit example is presented with Lee-Yang fusion rules. We briefly discuss applications to amalgamated crossed products, doubles, and quantum chains.

AB - By weakening the counit and antipode axioms of a C*-Hopf algebra and allowing for the coassociative coproduct to be nonunital, we obtain a quantum group, that we call a weak C*-Hopf algebra, which is sufficiently general to describe the symmetries of essentially arbitrary fusion rules. This amounts to generalizing the Baaj-Skandalis multiplicative unitaries to multipicative partial isometrics. Every finite-dimensional weak C*-Hopf algebra has a dual which is again a weak C*-Hopf algebra. An explicit example is presented with Lee-Yang fusion rules. We briefly discuss applications to amalgamated crossed products, doubles, and quantum chains.

KW - Amalgamated crossed products

KW - Multiplicative isometrics

KW - Weak Hopf algebras

UR - http://www.scopus.com/inward/record.url?scp=1542743743&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=1542743743&partnerID=8YFLogxK

M3 - Article

AN - SCOPUS:1542743743

VL - 38

SP - 437

EP - 456

JO - Letters in Mathematical Physics

JF - Letters in Mathematical Physics

SN - 0377-9017

IS - 4

ER -