Endomorphisms of Quantum Generalized Weyl Algebras

We prove that every endomorphism of a simple quantum generalized Weyl algebra A over a commutative Laurent polynomial ring in one variable is an automorphism. This is achieved by obtaining an explicit classification of all endomorphisms of A. Our main result applies to minimal primitive factors of the quantized enveloping algebra ({U_q({mathfrak{sl}}_2)}) and certain minimal primitive quotients of the positive part of ({U_q({mathfrak{so}}_5)}) .