  This is a survey of recent results on classification of compact quantum
groups of Lie type, by which we mean quantum groups with the same fusion rules
and dimensions of representations as for a compact connected Lie group $G$. The
classification is based on a categorical duality for quantum group actions
recently developed by De Commer and the authors in the spirit of Woronowicz's
Tannaka--Krein duality theorem. The duality establishes a correspondence
between the actions of a compact quantum group $H$ on unital C$^*$-algebras and
the module categories over its representation category Rep $H$. This is further
refined to a correspondence between the braided-commutative Yetter--Drinfeld
$H$-algebras and the tensor functors from Rep $H$. Combined with the more
analytical theory of Poisson boundaries, this leads to a classification of
dimension-preserving fiber functors on the representation category of any
coamenable compact quantum group in terms of its maximal Kac quantum subgroup,
which is the maximal torus for the $q$-deformation of $G$ if $q\ne1$. Together
with earlier results on autoequivalences of the categories Rep $G_q$, this
allows us to classify up to isomorphism a large class of quantum groups of
$G$-type for compact connected simple Lie groups $G$. In the case of $G=SU(n)$
this class exhausts all non-Kac quantum groups.

Quantum Algebra;Operator Algebras