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We introduce the notion of identity component of a compact quantum group andthat of total disconnectedness. As a drawback of the generalized Burnsideproblem, we note that totally disconnected compact matrix quantum groups mayfail to be profinite. We consider the problem of approximating the identitycomponent as well as the maximal normal (in the sense of Wang) connectedsubgroup by introducing canonical, but possibly transfinite, sequences ofsubgroups. These sequences have a trivial behaviour in the classical case. Wegive examples, arising as free products, where the identity component is notnormal and the associated sequence has length 1.  We give necessary and sufficient conditions for normality of the identitycomponent and finiteness or profiniteness of the quantum component group. Amongthem, we introduce an ascending chain condition on the representation ring,called Lie property, which characterizes Lie groups in the commutative case andreduces to group Noetherianity of the dual in the cocommutative case. It isweaker than ring Noetherianity but ensures existence of a generatingrepresentation. The Lie property and ring Noetherianity are inherited byquotient quantum groups. We show that A_u(F) is not of Lie type. We discuss anexample arising from the compact real form of U_q(sl_2) for q<0.

Connected components of compact matrix quantum groups and finiteness conditions