Connected components of compact matrix quantum groups and finiteness conditions

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 approximating the identity component as well as the maximal normal connected subgroup by introducing canonical, transfinite, sequences of subgroups, which have a trivial behaviour in the classical case. We give examples, arising as free products, where the identity component is not normal, in the sense of Wang, and the associated sequence has length $1$. We give necessary and sufficient conditions for normality of the identity component 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 commutative case and reduces to group Noetherianity of the 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 $A_u(F)$ is not of Lie type. We discuss an example arising from the compact real form of $U_q(\mathfrak{sl}_2)$ for $q<0$.