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

Abstract

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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.

How to cite

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Böhm, 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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  1. [1] G. Böhm, F. Nill, K. Szlachányi, Weak Hopf Algebras, in preparation. 
  2. [2] S. Baaj, G. Skandalis, Ann. Sci. ENS 26, 425 (1993). 
  3. [3] G. Böhm, K. Szlachányi, A Coassociative C*-Quantum Group with Non-Integral Dimensions, q-alq/9509008, Lett. Math. Phys. 35, 437 (1996). 
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  8. [8] R. Haag, Local Quantum Physics, Springer 1992. Zbl0777.46037
  9. [9] G. Mack and V. Schomerus, Endomorphisms and Quantum Symmetry of the Conformal Ising Model, in Algebraic Theory of Superselection Sectors, ed.: D. Kastler, World Scientific Singapore 1990. Zbl0715.17028
  10. [10] G. Mack and V. Schomerus, Nucl. Phys. B370, 185 (1992). 
  11. [11] F. Nill, K. Szlachányi, H.-W. Wiesbrock, in preparation. 
  12. [12] A. Ocneanu, Quantum Cohomology, Quantum Groupoids, and Subfactors, talk presented at the First Caribic School of Mathematics and Theoretical Physics, Guadeloupe 1993 (unpublished). 
  13. [13] K.-H. Rehren, Braid Group Statistics and their Superselection Rules in: Algebraic Theory of Superselection Sectors, ed. D. Kastler, World Scientific 1990. 
  14. [14] V. Schomerus, Quantum symmetry in quantum theory, DESY 93-018 preprint. 
  15. [15] M. E. Sweedler, Hopf algebras, Benjamin 1969. 
  16. [16] P. Vecsernyés, Nucl. Phys. B 415, 557 (1994). 
  17. [17] S. L. Woronowicz, Commun. Math. Phys. 111, 613 (1987). 

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