Automorphisms of Generalized Down-Up Algebras

TL;DR

This paper characterizes the automorphism groups of a broad class of generalized down-up algebras, extending previous work and applying results to specific Noetherian down-up algebras with certain root conditions.

Contribution

It explicitly describes the automorphism groups of conformal Noetherian generalized down-up algebras and applies these findings to classify automorphisms of specific Noetherian down-up algebras.

Findings

Automorphism groups explicitly described for conformal Noetherian generalized down-up algebras.

Application of results to characterize automorphisms of certain Noetherian down-up algebras.

Automorphism groups depend on roots of associated polynomials, with specific conditions on roots.

Abstract

A generalization of down-up algebras was introduced by Cassidy and Shelton (J. Algebra 279 (2004), no. 1), the so-called generalized down-up algebras. We describe the automorphism group of conformal Noetherian generalized down-up algebras L(f,r,s,\gamma) such that r is not a root of unity, listing explicitly the elements of the group. In the last section we apply these results to Noetherian down-up algebras, thus obtaining a characterization of the automorphism group of Noetherian down-up algebras A(\alpha, \beta, \gamma) for which the roots of the polynomial X^2-\alpha X-\beta are not both roots of unity.