  We classify up to isomorphism the quantum generalized Weyl algebras and
determine their automorphism groups in all cases in a uniform way, including
those where the parameter q is a root of unity, thereby completing the results
obtained by [Bavula, V. V.; Jordan, D. A. Isomorphism problems and groups of
automorphisms for generalized Weyl algebras. Trans. Amer. Math. Soc. 353
(2001), no. 2, 769-794] and [Richard, L.; Solotar, A. Isomorphisms between
quantum generalized Weyl algebras. J. Algebra Appl. 5 (2006), no. 3, 271-285]

Rings and Algebras