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Abstract. 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 (sl 2 ) and certain minimal primitive quotients of the positive part of U q (so 5 ).Mathematics Subject Classification (2010). 16W35, 16S32, 16W20, 17B37.

Motivated by Weyl algebra analogues of the Jacobian conjecture and the tame generators problem, we prove quantum versions of these problems for a family of analogues to the Weyl algebras. In particular, our results cover the Weyl-Hayashi algebras and tensor powers of a quantization of the first Weyl algebra which arises as a primitive factor algebra of U + q (so 5 ).Mathematics Subject Classification (2010). 16W35, 16S32, 16W20, 17B37.

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In this paper, we classify the derivations of degree-one generalized Weyl algebras over a univariate Laurent polynomial ring. In particular, our results cover the Weyl–Hayashi algebra, a quantization of the first Weyl algebra arising as a primitive factor algebra of [Formula: see text], and a family of algebras which localize to the group algebra of the infinite group with generators [Formula: see text] and [Formula: see text], subject to the relation [Formula: see text]