Abstract
Some examples of quantum groups in literature arise as deformations of a locally compact group by a “dual” 2-cocycle. We make this construction in the framework of Kac algebras; we show that these deformations are still Kac algebras; using this construction, we give new quantizations of the Heisenberg group. From this point of view, we analyse the dimension 8 non-trivial example of Kac and Paljutkin, and give a new example of non-trivial dimension 12 semi-simple *-Hopf algebras (a dimension 12 Kac algebra).
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[BS] Baaj, S., Skandalis, G.: Unitaires multiplicatifs et dualité pour les produits croisés de ℂ*-algébres. Ann. Sci. ENS26, 425–488 (1993)
[D] Drinfeld, V.G.: Quantum groups. Proceedings ICM Berkeley (1986), vol.1, pp. 798–820
[ER] Effros, E.G., Ruan, Z.-J.: Discrete quantum groups, I; the Haar measure. Int. J. Math.5, 681–723 (1994)
[ES1] Enock, M., Schwartz, J.-M.: Algèbres de Kac moyennables. Pacific J. Math.125, 363–379 (1986)
[ES2] Enock, M., Schwartz, J.-M.: Kac algebras and duality of locally compact groups. Berlin: Springer, 1992
[EV] Enock, M., Vallin, J.-M.: ℂ*-algébres de Kac et algèbres de Kac. Proc. London Math. Soc.66, 619–650 (1993)
[KP1] Кац, Г.И., Палюткин, В.Г. (G.I. Kac, V.G. Paljutkin): Пример кольцевой группь, порожденной группами Ли.Укр. Мамем. Ж. 16, 99–105 (1964)
[KP2] Кац, Г.И., Палюткин, В.Г. (G.I. Kac, V.G. Paljutkin): Пример кольцевой группы восьмого порядка.Ycnexu Mam. Hayk,20–5, 268–269 (1965)
[KP3] Кац, Г.И., Палюткин, В.Г. (G.I. Kac, V.G. Paljutkin): Конечные кольцевые группы.Трубы Моск. Мамем. об-еа 15, 224–261 (1966); Translated in: Finite ring-groups. Trans. Moscow Math. Soc. 251–294 (1966)
[L] Landstad, M.B.: Quantization arising from abelian subgroups. Int. J. Math.5, 897–936 (1994)
[LR] Landstad, M.B., Raeburn, I.R.: Twisted dual-group algebras: Equivariant deformations ofC 0(G). J. Funct. Anal.132, 43–85 (1995)
[L'R'] Larson, R.G., Radford, D.E.: Semisimple Hopf algebras. J. Alg.171, 5–35 (1995)
[MN] Masuda, T., Nakagami, Y.: A von Neumann Algebra framework for the duality of the quantum groups. Publ. RIMS Kyoto,30, 799–850 (1994)
[PW] Podleś, P., Woronowicz, S.L.: Quantum Deformation of Lorentz Group. Commun. Math. Phys.130, 381–431 (1990)
[R1] Rieffel, M.A.: Compact Quantum groups associated with Toral subgroups. Contemp. Math.145, 465–491 (1993)
[R2] Rieffel, M.A.: Deformation Quantization for Actions of ℝd. Memoirs A.M.S.,506 (1993)
[TT] Takesaki, M., Tatsuuma, N.: Duality and subgroups. Ann. Math.93, 344–364 (1971)
[V] Van Daele, A.: Quantum deformations of the Heisenberg group. Proc. Satellite Conference of ICM-90, Singapore: World Sci. 1991, pp. 314–325
[Wa] Wang, S.: Deformation of Compact Quantum Groups via Rieffel's Quantization, to appear in Commun. Math. Phys.
[Wi] Williams, R.: Finite dimensional Hopf algebras. Thesis, Florida State University, 1988
[W1] Woronowicz, S.L.: Tannaka-Krein duality for compact matrix pseudogroups. TwistedSU(N) group. Invent. Math.93, 35–76 (1988)
[W2] Woronowicz, S.L.: Compact quantum groups. Preprint 1993
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Communicated by A. Connes
Dedicated to the memory of George Kac ( )
Research of the second author was supported in part by the Ukrainian Foundation for Fundamental Studies and by the International Science Foundation.
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Enock, M., Vaînerman, L. Deformation of a Kac algebra by an abelian subgroup. Commun.Math. Phys. 178, 571–595 (1996). https://doi.org/10.1007/BF02108816
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DOI: https://doi.org/10.1007/BF02108816