Quantum ax + b Group as Quantum Automorphism Group of k[x]

By introducing a result that guarantees a given bialgebra to be a Hopf algebra under a natural condition, we show that the quantum automorphism group of the algebra k[x] of polynomials over a field k (of any characteristic) is the universal quantum a x + b group, generalizing the fact that the automorphism group of k[x] is the a x + b group. The q-deformation of the a x + b group is then seen as one among a certain family ${\mathcal{A}}_{q,n}$ of quantum subgroups of this universal quantum group.