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CQG algebras: A direct algebraic approach to compact quantum groups. (English) Zbl 0861.17005

In this nice article, it is explained what algebraic data are necessary to construct a compact quantum group. The main ideas are in the spirit of previous papers [S. L. Woronowicz, Invent. Math. 93, 35-76 (1988; Zbl 0644.58044); M. Rosso, Duke Math. J. 61, No. 1, 11-40 (1990; Zbl 0721.17013), N. Andruskiewitsch, Bull. Soc. Math. Fr. 120, No. 3, 297-325 (1992; Zbl 0763.17008)]. However there is a key step in passing to the \(C^*\)-algebra setting which was implicit in those previous papers and is proved here in detail. (Independently, an analogous proof was communicated by the reviewer to A. Guichardet and appeared in his [Groupes quantiques, Inter Éditions/CNRS Editions (1995; Zbl 0838.17007)]). Similar presentations are also given in [Shuzhou Wang, Dissertation, Berkeley (1993), E. G. Effros and Z.-J. Ruan, Int. J. Math. 5, No. 5, 681-723 (1994; Zbl 0824.17020)] – as noted by the authors.

MSC:

17B37 Quantum groups (quantized enveloping algebras) and related deformations
16W30 Hopf algebras (associative rings and algebras) (MSC2000)
46L89 Other “noncommutative” mathematics based on \(C^*\)-algebra theory
22C05 Compact groups
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References:

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