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Examples of locally compact quantum groups through the bicrossed product construction. (English) Zbl 1045.46042
Grigoryan, A. (ed.) et al., XIIIth international congress on mathematical physics. Proceedings of the ICMP 2000, Imperial College, London, Great Britain, July 17–22, 2000. Boston, MA: International Press (ISBN 1-57146-085-3). 341-348 (2001).
Compact groups form a category closed under duality, and every group in the category is canonically isomorphic to its bidual: this is the Pontryagin–van Kampen theorem. To generalize this to locally compact groups requires enlarging the category of groups to that of Kac algebras [L. I. Vajnerman and G. I. Kac, Math. USSR, Sb. 23 (1974), 185–214 (1975; Zbl 0309.46052); M. Enock and J.-M. Schwartz, “Kac algebras and duality of locally compact groups” (Springer, Berlin) (1992; Zbl 0805.22003)]. If the category is also to include the locally compact quantum groups of Woronowicz, it must be further enlarged, and the objects are termed locally compact quantum groups (lcq-groups) [J. Kustermans and S. Vaes, Ann. Sci. Éc. Norm. Supér., IV. Sér. 33, 837–934 (2000; Zbl 1034.46508)]. In the words of the author, the main aim of this paper is “to introduce some examples of non-compact quantum groups to a non-specialized audience.”
The author begins with the argument outlined above, but in more detail. He then points out that lcq-groups arise in a natural way as symmetries of quantum spaces, and that they lead to a topological version of a Hopf algebra.
In describing examples of lcq-groups, the author relies on the definition based on a von Neumann algebra \(M\) equipped with a comultiplication \(\Delta: M\to M\otimes M\) and two normal semi-finite faithful left and right invariant weights, respectively, on \(M\) (analogues of Haar measure); the antipode is then a derived quantity. (It is the hypothesis of the existence of these weights that is crucial to this construction.)
The first example recovers familiar ground, now in this formulation: \(M\) is the algebra \(L^\infty(G)\) for any locally compact group \(G\), and the weights are integration with respect to left and right Haar measure. The dual to \(M\) is \(\mathcal{L}(G)\), the group von Neumann algebra.
The next examples mentioned are due to Woronowicz, Baaj and Van Daele: the quantum group versions of \(E(2)\), \(x\mapsto ax+b\), \(z\mapsto az+b\). References are given, but details are omitted due to the complexity of the constructions.
Then comes the main example: lcq-groups through bicrossed products. For this one needs the notion of matched pairs \((G,H)\) of locally compact groups. The basic von Neumann algebra \(M\) is that generated by \(L^\infty(H)\) mapped into \(L^\infty(G\times H)\) by a \(G\)-action, and \(\mathcal{L}(G)\otimes I\). This \(M\) acts on \(L^2(G\times H)\). In the final section, the author details the special case where \(G\) is the additive group of \(\mathbb{R}\) and \(H\) is \(\mathbb{R}^+_*\times \mathbb{R}\) equipped with the group law \((a,b)\cdot (c,d)=(ac,ad+b/c)\). It turns out that \(M\) is the universal enveloping algebra of the Heisenberg Lie algebra. The dual lcq-group is obtained by interchanging \(G\) and \(H\), in which case the “co-algebra structure precisely reflects the structure of the Heisenberg group \(\ldots\) This is typical for the duality of quantum groups.”
For the entire collection see [Zbl 0994.00023].

46L65 Quantizations, deformations for selfadjoint operator algebras
20G42 Quantum groups (quantized function algebras) and their representations
58B32 Geometry of quantum groups
81R50 Quantum groups and related algebraic methods applied to problems in quantum theory
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