Multiple noncommutative tori as quantum symmetry groups

• Published in: Journal of geometry and physics Vol. 131; pp. 1 - 22
• Main Authors: Banacki, Michał, Marciniak, Marcin
• Format: Journal Article
• Language: English
• Published: Elsevier B.V 01.09.2018
• Subjects: Compact quantum group action; Compact quantum groups; Hopf algebra; Multiple noncommutative torus; Noncommutative torus; Woronowicz C-algebra; secondary; Multiple noncommutative torus; Compact quantum groups; Compact quantum group action; Hopf algebra; Woronowicz C-algebra; primary; Noncommutative torus
• ISSN: 0393-0440; 1879-1662
• DOI: 10.1016/j.geomphys.2018.04.010

We discuss necessary conditions for a compact quantum group to act on the algebra of noncommutative n-torus Tθn in a filtration preserving way in the sense of Banica and Skalski. As a result, we construct a family of compact quantum groups Gθ=(Aθn,Δ) such that for each θ, Gθ is the final object in the category of all compact quantum groups acting on Tθn in a filtration preserving way. We describe in detail the structure of the C*-algebra Aθn and provide a concrete example of its representation in bounded operators. Moreover, we show that Aθn is isomorphic to the C*-completed version of an algebra of multiple noncommutative torus. Finally, we compute the Haar measure of Gθ and discuss its representation theory. For θ=0, the quantum group G0 is nothing but the classical group Tn⋊Sn, where Sn is the symmetric group.