What is a quantum group? What field of mathematics does the study of quantum groups belong to? Quantum groups are studied by many mathematicians including topologists (because of their links to knot theory), mathematical physicists (because of their link to quantum theory and field theory, including the quantum inverse scattering method and the Yang-Baxter equation), geometers (because of their connection to non-commutative geometry), workers in Lie theory and algebraic groups (because of their connection to the study of algebraic groups in positive characterstic), and ring theorists (because they are rings -- Hopf algebras in fact).
A quantum group is not a group! It is the (generally non-commutative) ring of "functions" on something like a group. It is a Hopf algebra probably with some extra structure (some sort of triangular or co-triangular structure?); in analytic studies, it may be a C*-algebra.
The members of the algebra group who work in this area are Ian Musson and Yi Ming Zou (with Allen Bell having some interests along this line).