The automorphism groups for a family of generalized Weyl algebras

In this paper, we study a family of generalized Weyl algebras {A(p) (lambda, mu, K-q[s, t])} On and their polynomial extensions. We will show that the algebra A(p) (lambda, mu, K-q[s, t]) has a simple localization A p A(p) (lambda, mu, K-q[s, t])s when none of p and q is a root of unity. As an application, we determine all the height-one prime ideals and the center for A(p) (lambda, mu, K-q[s, t]), and prove that A(p) (lambda, mu, K-q[s, t]) is cancellative. Then we will determine the automorphism group and solve the isomorphism problem for the generalized Weyl algebras A(p) (lambda, mu, K-q[s, t]) and their polynomial extensions in the case where none of p and q is a root. of unity. We will establish a quantum analogue of the Dixmier conjecture and compute the automorphism group for the simple localization (A(p) (1, 1, K-q[s, t]))s. Moreover, we will completely determine the automorphism group for the algebra A(p) (1, 1, K-q[s, t]), and its polynomial extension when p not equal 1 and q not equal 1.