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**Discrete quantum groups. I: The Haar measure.**
*(English)*
Zbl 0824.17020

The authors introduce the notion of a discrete quantum group and prove the existence of the Haar measure. Compact quantum groups are now reasonably well understood. They have been developed by S. L. Woronowicz [see Commun. Math. Phys. 111, 613-665 (1987; Zbl 0627.58034) and Compact quantum groups, Preprint Univ. Warsaw (1993)]. In [Commun. Math. Phys. 130, 381-431 (1990; Zbl 0703.22018)], P. Podleś and S. L. Woronowicz have studied discrete quantum groups as duals of compact quantum groups. The authors of the paper under review are the first to study discrete quantum groups as independent objects. They define a discrete quantum group as a cosemisimple Hopf *-algebra with a standard *-operation. This means that they consider Hopf *-algebras such that the dual *-algebra is a direct product of full matrix algebras over the complex numbers with the usual *-algebra structure.

An extensive part of the paper is used to develop the necessary tools: cosemisimple Hopf algebras, representation theory, the multimatrix algebra structure of the dual algebra, intertwining formulas for certain representations, …The main result however is the existence of the left and right Haar measures. Also the duality between the discrete and the compact quantum groups is discussed.

This work relates to many other different approaches to discrete and compact quantum groups. We have mentioned already the work of Podleś and Woronowicz. It is no surprise that essentially the same formulas are obtained. There is also the work of T. Koornwinder and M. Dijkhuizen [CQG algebras: A direct algebraic approach to compact quantum groups, Lett. Math. Phys. 32, 315-330 (1994)]. They give a purely algebraic definition of compact quantum groups by saying that it is a Hopf *-algebra that is spanned by the matrix elements of the finite- dimensional unitary corepresentations. Such a notion was also given by E. Kirchberg during a meeting in Oberwolfach in 1994, but he called them discrete quantum groups. All these objects are essentially the same. The confusion comes from the fact that there are essentially two choices one can make: either one takes the group algebra (with convolution product) as the basic object to deform, or one takes the function algebra (with pointwise product).

Finally, there is also the reviewer’s paper [Discrete quantum groups, Preprint, Univ. Leuven (1993), to appear in J. Algebra], where the discrete quantum groups are studied as multiplier Hopf algebra [the reviewer, Trans. Am. Math. Soc. 342, 917-932 (1994; Zbl 0809.16047)] and where an easy proof of the existence of the Haar measure is given.

An extensive part of the paper is used to develop the necessary tools: cosemisimple Hopf algebras, representation theory, the multimatrix algebra structure of the dual algebra, intertwining formulas for certain representations, …The main result however is the existence of the left and right Haar measures. Also the duality between the discrete and the compact quantum groups is discussed.

This work relates to many other different approaches to discrete and compact quantum groups. We have mentioned already the work of Podleś and Woronowicz. It is no surprise that essentially the same formulas are obtained. There is also the work of T. Koornwinder and M. Dijkhuizen [CQG algebras: A direct algebraic approach to compact quantum groups, Lett. Math. Phys. 32, 315-330 (1994)]. They give a purely algebraic definition of compact quantum groups by saying that it is a Hopf *-algebra that is spanned by the matrix elements of the finite- dimensional unitary corepresentations. Such a notion was also given by E. Kirchberg during a meeting in Oberwolfach in 1994, but he called them discrete quantum groups. All these objects are essentially the same. The confusion comes from the fact that there are essentially two choices one can make: either one takes the group algebra (with convolution product) as the basic object to deform, or one takes the function algebra (with pointwise product).

Finally, there is also the reviewer’s paper [Discrete quantum groups, Preprint, Univ. Leuven (1993), to appear in J. Algebra], where the discrete quantum groups are studied as multiplier Hopf algebra [the reviewer, Trans. Am. Math. Soc. 342, 917-932 (1994; Zbl 0809.16047)] and where an easy proof of the existence of the Haar measure is given.

Reviewer: A.Van Daele (Heverlee)