Automorphisms of Generalized Down-Up Algebras

Abstract

A generalization of down-up algebras was introduced by Cassidy and Shelton in [11], the so-called generalized down-up algebras. We describe the automorphism group of conformal Noetherian generalized down-up algebras L(f, r, s, γ) 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(α, β, γ) for which the roots of the polynomial X 2 − αX − β are not both roots of unity.

Classification of factorial generalized down–up algebras

“…In order to discuss the existence of principal eigenvectors, we will make use of two integers ǫ ∈ Z and τ ∈ N, which have been defined in [9], as follows: Proof. By Proposition 4.1, we can work over the localisation L h .…”

Primitive Ideals of Noetherian Generalized Down-Up Algebras

“…We will, however, continue to use γ and φ without additional assumptions, since assuming γ = 1 or φ monic does not significantly ease our workload. But there is an isomorphism between generalized down-up algebras that we will exploit heavily; see [CL,Proposition 1.7] Lemma 1.1. Suppose r = 1.…”

“…To determine exactly when an algebra is conformal is not a complete triviality, mostly due to the presence of γ, but Lemma 1.1 allows us to ignore γ most of the time, so we can determine quickly which of our algebras are conformal [CL,Lemma 1.6,Proposition 1.8].…”

Twisted Generalized Weyl Algebras, Polynomial Cartan Matrices and Serre-Type Relations

“…Their structure and representation theory have been extensively studied in varying degrees of generality, see [5], [6], [3], [4], [8], [10] and references therein. Examples of generalized Weyl algebras include the n:th Weyl algebra A n , the enveloping algebra U(sl 2 ) and the quantum group U q (sl 2 ) as well as many other interesting families of algebras (see for example [7] and references therein).…”