We introduce the notion of identity component of a compact quantum group and that of total disconnectedness. As a drawback of the generalized Burnside problem, we note that totally disconnected compact matrix quantum groups may fail to be profinite. We consider the problem of constructing the identity component by introducing canonical approximating transfinite sequences of subgroups. These sequences have lengths ≤1 in the classical case but can be countably infinite for duals of discrete groups. We give examples of free product quantum groups where the identity component is not normal and the associated sequence has length 1.We give necessary and sufficient conditions for normality of the identity component, in the sense of Wang, and finiteness or profiniteness of the quantum component group. Among them, we introduce an ascending chain condition on the representation ring, called Lie property, which characterizes Lie groups in the classical case and reduces to group Noetherianity of the discrete group dual in the cocommutative case. It is weaker than ring Noetherianity but ensures existence of a generating representation. The Lie property and ring Noetherianity are inherited by quotient quantum groups. We show that Au(F) is not of Lie type. We discuss an example arising from the compact real form of Uq(sl2) for q<0.

Cirio L.S., D'Andrea A., Pinzari C., Rossi S. (2014). Connected components of compact matrix quantum groups and finiteness conditions. JOURNAL OF FUNCTIONAL ANALYSIS, 267(9), 3154-3204 [10.1016/j.jfa.2014.08.022].

Connected components of compact matrix quantum groups and finiteness conditions

D'Andrea A.;
2014

Abstract

We introduce the notion of identity component of a compact quantum group and that of total disconnectedness. As a drawback of the generalized Burnside problem, we note that totally disconnected compact matrix quantum groups may fail to be profinite. We consider the problem of constructing the identity component by introducing canonical approximating transfinite sequences of subgroups. These sequences have lengths ≤1 in the classical case but can be countably infinite for duals of discrete groups. We give examples of free product quantum groups where the identity component is not normal and the associated sequence has length 1.We give necessary and sufficient conditions for normality of the identity component, in the sense of Wang, and finiteness or profiniteness of the quantum component group. Among them, we introduce an ascending chain condition on the representation ring, called Lie property, which characterizes Lie groups in the classical case and reduces to group Noetherianity of the discrete group dual in the cocommutative case. It is weaker than ring Noetherianity but ensures existence of a generating representation. The Lie property and ring Noetherianity are inherited by quotient quantum groups. We show that Au(F) is not of Lie type. We discuss an example arising from the compact real form of Uq(sl2) for q<0.
2014
Cirio L.S., D'Andrea A., Pinzari C., Rossi S. (2014). Connected components of compact matrix quantum groups and finiteness conditions. JOURNAL OF FUNCTIONAL ANALYSIS, 267(9), 3154-3204 [10.1016/j.jfa.2014.08.022].
Cirio L.S.; D'Andrea A.; Pinzari C.; Rossi S.
File in questo prodotto:
Eventuali allegati, non sono esposti

I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.

Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11585/952702
 Attenzione

Attenzione! I dati visualizzati non sono stati sottoposti a validazione da parte dell'ateneo

Citazioni
  • ???jsp.display-item.citation.pmc??? ND
  • Scopus 14
  • ???jsp.display-item.citation.isi??? 8
social impact