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].

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].

Reviewer: Daniel A. Dubin (Oxford)

##### MSC:

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 |