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**Double group construction of quantum groups in the von Neumann algebra framework.**
*(English)*
Zbl 0998.46040

The author performs V. G. Drinfeld’s double construction [see Proc. Int. Congr. Math. Berkeley, Calif., 1986, Vol. 1, 798-820 (1987; Zbl 0667.16003)] in the framework of Woronowicz algebras. The concept of a Woronowicz algebra was introduced by T. Masuda and Y. Nakagami in [Publ. Res. Inst. Math. Sci. 30, No. 5, 799-850 (1994; Zbl 0839.46055)] in order to deal with quantum groups in the von Neumann algebraic setting. The definition of a Woronowicz algebra is quite complicated: the starting point is a von Neumann algebra with comultiplication and thereon, one should have an antipode with polar decomposition, a Haar measure and several relations between these different objects. Because there is a gap in the work of Masuda and Nakagami, it was not clear for the author that the dual of a Woronowicz algebra is again a Woronowicz algebra. Therefore, he introduces the notion of a quasi Woronowicz algebra, which only differs from a Woronowicz algebra in a technical aspect.

This class of quasi Woronowicz algebras is self-dual and for them, the author defines the quantum double. As a von Neumann algebra with comultiplication, the quantum double is easy to describe: the tensor product comultiplication on the tensor product of a quasi Woronowicz algebra and its dual is deformed by a natural twist. Then, the author computes the Haar measure and the antipode with polar decomposition and proves that he gets again a quasi Woronowicz algebra. It is also shown that the associated multiplicative unitary agrees with the quantum double multiplicative unitary introduced by S. Baaj and G. Skandalis in [Ann. Sci. Ëc. Norm. Supér., IV. Ser. 26, No. 4, 425-488 (1993; Zbl 0804.46078)].

Recently J. Kustermans and the reviewer have given a simpler definition of locally compact quantum groups, both in the C\(^*\)-algebraic language [see Ann. Sci. Éc. Norm. Supér., IV. Sér. 33, No. 6, 837-934 (2000; Zbl 1034.46508) and C. R. Acad. Sci. Paris Sér. I Math. 328, No. 10, 871-876 (1999; Zbl 0957.46037)]) and the von Neumann algebraic language [see J. Kustermans and S. Vaes, Locally compact quantum groups in the von Neumann algebraic setting. Math. Scand., to appear]. In this setting, it is easier to obtain the quantum double, because less axioms have to be verified. It also follows from these papers that every quasi Woronowicz algebra is in fact a Woronowicz algebra.

This class of quasi Woronowicz algebras is self-dual and for them, the author defines the quantum double. As a von Neumann algebra with comultiplication, the quantum double is easy to describe: the tensor product comultiplication on the tensor product of a quasi Woronowicz algebra and its dual is deformed by a natural twist. Then, the author computes the Haar measure and the antipode with polar decomposition and proves that he gets again a quasi Woronowicz algebra. It is also shown that the associated multiplicative unitary agrees with the quantum double multiplicative unitary introduced by S. Baaj and G. Skandalis in [Ann. Sci. Ëc. Norm. Supér., IV. Ser. 26, No. 4, 425-488 (1993; Zbl 0804.46078)].

Recently J. Kustermans and the reviewer have given a simpler definition of locally compact quantum groups, both in the C\(^*\)-algebraic language [see Ann. Sci. Éc. Norm. Supér., IV. Sér. 33, No. 6, 837-934 (2000; Zbl 1034.46508) and C. R. Acad. Sci. Paris Sér. I Math. 328, No. 10, 871-876 (1999; Zbl 0957.46037)]) and the von Neumann algebraic language [see J. Kustermans and S. Vaes, Locally compact quantum groups in the von Neumann algebraic setting. Math. Scand., to appear]. In this setting, it is easier to obtain the quantum double, because less axioms have to be verified. It also follows from these papers that every quasi Woronowicz algebra is in fact a Woronowicz algebra.

Reviewer: Stefaan Vaes (Leuven)

### MSC:

46L65 | Quantizations, deformations for selfadjoint operator algebras |

81R50 | Quantum groups and related algebraic methods applied to problems in quantum theory |

22D25 | \(C^*\)-algebras and \(W^*\)-algebras in relation to group representations |

46L10 | General theory of von Neumann algebras |