On the automorphisms of $U_q^+({\mathfrak g})$

Abstract

Let g be a simple complex finite dimensional Lie algebra and let U_q^+(g) be the positive part of the quantum enveloping algebra of g. If g is of type A_2, the group of algebra automorphisms of U_q^+(g) is a semidirect product of a 2-dimensional torus and the group of automorphisms of the Dynkin diagram; any algebra automorphism is an automorphism of braided Hopf algebra, and preserves the standard grading [AD1]. 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 B_2. We study Hopf algebra automorphisms of Nichols algebras and their bosonizations and compute in particular the group of Hopf algebra automorphisms of U_q^+(g).