Let g be a simple complex finite dimensional Lie algebra and let Uq+(ℊ) be the positive part of the quantum enveloping algebra of g.If g is of type A2, the group of algebra automorphisms of Uq+(ℊ) is a semidirect product (k×)2 ⋊ Autdiagr(ℊ) any algebra automorphism is an automorphism of braided Hopf algebra and preserves the standard grading . This intriguing smallness of the group of algebra automorphisms raises questions about the extent of these phenomena. We discuss some of them in the present paper. We introduce the notion of “algebra with few automorphisms” and establish some consequences. We prove some exploratory results concerning the group of algebra automorphisms for the type B2. We study the Hopf algebra automorphisms of Nichols algebras and their bosonizations and compute in particular the group of Hopf algebra automorphisms of Uq+(ℊ).