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

• Autorzy: Gabriella Böhm , Kornél Szlachányi
• Języki publikacji: EN

Abstrakty

EN

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.

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Banach Center Publications
• Rocznik: 1997
• Tom: 40
• Numer: 1
• Strony: 9-19

Daty

wydano

1997

Twórcy

autor

Gabriella Böhm

• Central Research Institute for Physics, H-1525 Budapest 114, P.O.B. 49, Hungary

autor

Kornél Szlachányi

• Central Research Institute for Physics, H-1525 Budapest 114, P.O.B. 49, Hungary

Bibliografia

• [1] G. Böhm, F. Nill, K. Szlachányi, Weak Hopf Algebras, in preparation.
• [2] S. Baaj, G. Skandalis, Ann. Sci. ENS 26, 425 (1993).
• [3] G. Böhm, K. Szlachányi, A Coassociative C*-Quantum Group with Non-Integral Dimensions, q-alq/9509008, Lett. Math. Phys. 35, 437 (1996).
• [4] S. Doplicher, J. E. Roberts, Ann. Math. 130 75 (1989), and Commun. Math. Phys. 131, 51 (1990).
• [5] V. G. Drinfeld, Leningrad Math. J. 1 1419, (1990).
• [6] K. Fredenhagen, K.-H. Rehren and B. Schroer, Commun. Math. Phys. 125, 201 (1989).
• [7] Fuchs, Ganchev, Vecsernyés, Towards a classification of rational Hopf algebras, preprint NIKHEF-H/94-05 KL-TH-94/4, hep-th 9402 153.
• [8] R. Haag, Local Quantum Physics, Springer 1992.
• [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.
• [10] G. Mack and V. Schomerus, Nucl. Phys. B370, 185 (1992).
• [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and Subfactors, talk presented at the First Caribic School of Mathematics and Theoretical Physics, Guadeloupe 1993 (unpublished).
• [13] K.-H. Rehren, Braid Group Statistics and their Superselection Rules in: Algebraic Theory of Superselection Sectors, ed. D. Kastler, World Scientific 1990.
• [14] V. Schomerus, Quantum symmetry in quantum theory, DESY 93-018 preprint.
• [15] M. E. Sweedler, Hopf algebras, Benjamin 1969.
• [16] P. Vecsernyés, Nucl. Phys. B 415, 557 (1994).
• [17] S. L. Woronowicz, Commun. Math. Phys. 111, 613 (1987).