QUANTUMax+bGROUP AS QUANTUM AUTOMORPHISM GROUP OFk[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 ax+b group , generalizing the fact that the automorphism group of k[x] is the ax+b group. The q-deformation of the ax+b group is then seen as one among a certain family (q ∈ k, n an integer) of quantum subgroups of this universal quantum group.