The Automorphism Groups for a family of Generalized Weyl Algebras

2016/10/14 by Xin Tang, Tang, Xin
Mathematics · #Algebraic structures and combinatorial models #Advanced Algebra and Geometry #Advanced Topics in Algebra

paper · pdf · doi:10.48550/arxiv.1610.04535

Abstract

In this paper, we study a family of generalized Weyl algebras \\A\ and their polynomial extensions. We will show that the algebra \A has a simple localization \A_\mathbbS 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, and prove that \A is cancellative. Then we will determine the automorphism group and solve the isomorphism problem for the generalized Weyl algebras \A 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]))_\mathbbS. 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≠ 1 and q≠ 1.

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