Hilbert's fifth problem inquires whether every locally Euclidean group is necessarily a Lie group. Von Neumann demonstrated that this is indeed true for the compact case.
The definition of a quantum group is not universally agreed upon. For some, it specifically refers to deformations of Lie groups, whereas for others, it encompasses the broader concept of a Hopf algebra.
Is there a quantum analogue of von Neumann's result? Specifically, under what conditions can a Hopf algebra be viewed as a deformation of a Lie group?
I am not aware of such a result, even if we restrict to quasitriangular Hopf algebras. Any insights or references on possible quantum analogues would be greatly appreciated.
An initial step could involve focusing on compact quantum groups and developing a non-commutative generalization of the concept of "locally Euclidean" that applies to them.
In contemporary terminology, the concept of a locally Euclidean space is understood as a topological manifold. One approach to the notion of noncommutative topological manifold is explored here; however, it is not compatible with the property of compactness.
