# Weak c*-Hopf algebras: the coassociative symmetry of non-integral dimensions

Gabriella Böhm; Kornél Szlachányi

Banach Center Publications (1997)

- Volume: 40, Issue: 1, page 9-19
- ISSN: 0137-6934

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topBöhm, Gabriella, and Szlachányi, Kornél. "Weak c*-Hopf algebras: the coassociative symmetry of non-integral dimensions." Banach Center Publications 40.1 (1997): 9-19. <http://eudml.org/doc/252204>.

@article{Böhm1997,

abstract = {By allowing the coproduct to be non-unital and weakening the counit and antipode axioms of a C*-Hopf algebra too, we obtain a selfdual set of axioms describing a coassociative quantum group, that we call a weak C*-Hopf algebra, which is sufficiently general to describe the symmetries of essentially arbitrary fusion rules. It is the same structure that can be obtained by replacing the multiplicative unitary of Baaj and Skandalis with a partial isometry. The algebraic properties, the existence of the Haar measure and representation theory are briefly discussed. An algorithm is explained how to construct examples (in particular ones with non-integral dimensions) from non-Abelian cohomology.},

author = {Böhm, Gabriella, Szlachányi, Kornél},

journal = {Banach Center Publications},

keywords = {weak -Hopf algebras; quantum groupoids; weak -algebras},

language = {eng},

number = {1},

pages = {9-19},

title = {Weak c*-Hopf algebras: the coassociative symmetry of non-integral dimensions},

url = {http://eudml.org/doc/252204},

volume = {40},

year = {1997},

}

TY - JOUR

AU - Böhm, Gabriella

AU - Szlachányi, Kornél

TI - Weak c*-Hopf algebras: the coassociative symmetry of non-integral dimensions

JO - Banach Center Publications

PY - 1997

VL - 40

IS - 1

SP - 9

EP - 19

AB - By allowing the coproduct to be non-unital and weakening the counit and antipode axioms of a C*-Hopf algebra too, we obtain a selfdual set of axioms describing a coassociative quantum group, that we call a weak C*-Hopf algebra, which is sufficiently general to describe the symmetries of essentially arbitrary fusion rules. It is the same structure that can be obtained by replacing the multiplicative unitary of Baaj and Skandalis with a partial isometry. The algebraic properties, the existence of the Haar measure and representation theory are briefly discussed. An algorithm is explained how to construct examples (in particular ones with non-integral dimensions) from non-Abelian cohomology.

LA - eng

KW - weak -Hopf algebras; quantum groupoids; weak -algebras

UR - http://eudml.org/doc/252204

ER -

## References

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- [11] F. Nill, K. Szlachányi, H.-W. Wiesbrock, in preparation.
- [12] A. Ocneanu, Quantum Cohomology, Quantum Groupoids, and Subfactors, talk presented at the First Caribic School of Mathematics and Theoretical Physics, Guadeloupe 1993 (unpublished).
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