Quantum \(E(2)\) group and its Pontryagin dual. (English) Zbl 0752.17017
This paper is concerned with the quantum group \(E_ \mu(2)\) [see also the author, Commun. Math. Phys. 136, 399-432 (1991; Zbl 0743.46080), L. Vaksman and L. Korogodskij, Dokl. Akad. Nauk SSSR 304, 1036- 1040 (1989; Zbl 0699.46053), E. Celeghini, R. Giachetti, E. Sorace and M. Tarlini, J. Math. Phys. 31, 2548-2551 (1990; Zbl 0725.17020)].
The author recalls the definition of the \(C^*\)-algebra of “continuous functions vanishing at infinity” on this quantum group and proves the existence of the comultiplication, in the operator-theory framework. Then he proves a theorem characterizing the unitary representations of \(E_ \mu(2)\), that is, the unitary comodules of that algebra. The same task is accomplished with the Pontryagin dual of \(E_ \mu(2)\), which is, in addition, shown to be a deformation of the group of transformations of the plane generated by translations and dilations.
The author recalls the definition of the \(C^*\)-algebra of “continuous functions vanishing at infinity” on this quantum group and proves the existence of the comultiplication, in the operator-theory framework. Then he proves a theorem characterizing the unitary representations of \(E_ \mu(2)\), that is, the unitary comodules of that algebra. The same task is accomplished with the Pontryagin dual of \(E_ \mu(2)\), which is, in addition, shown to be a deformation of the group of transformations of the plane generated by translations and dilations.
Reviewer: N.Andruskiewitsch (Bonn)
MSC:
| 17B37 | Quantum groups (quantized enveloping algebras) and related deformations |
| 22D25 | \(C^*\)-algebras and \(W^*\)-algebras in relation to group representations |
| 22D35 | Duality theorems for locally compact groups |
| 46L60 | Applications of selfadjoint operator algebras to physics |
| 81R50 | Quantum groups and related algebraic methods applied to problems in quantum theory |
Keywords:
quantum groups; non-compact quantum groups; \(C^*\)-algebra; comultiplication; unitary representations; Pontryagin dual; group of transformations; quantum deformation; two-fold covering of group motions; Euclidean planeReferences:
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