Samuel A. Lopes

Automorphisms of Generalized Down-Up Algebras

A generalization of down-up algebras was introduced by Cassidy and Shelton (2004), 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 − α X − β are not both roots of unity.

Primitive Ideals of

Let U q (𝔤+) be the quantized enveloping algebra of the nilpotent Lie algebra 𝔤+ = 𝔰𝔩n+1 + which occurs as the positive part in the triangular decomposition of the simple Lie algebra 𝔰𝔩 n+1 of type A n . Assuming the base field 𝕂 is algebraically closed and of characteristic 0, and that the parameter q ∈ 𝕂* is not a root of unity, we define and study certain quotients of U q (𝔤+) which coincide with the Weyl–Hayashi algebra when n = 2 (see Alev and Dumas, 1996; Hayashi, 1990; Kirkman and Small, 1993). We show that these are simple Noetherian domains, with a trivial center and even Gelfand-Kirillov dimension. Hence, they play a role analogous to that played by the Weyl algebras in the classical case. In the remainder of the article, we study the primitive spectrum of Uq (𝔰𝔩4 + in detail, somewhat in the spirit of Launois (to appear). We determine all primitive ideals of Uq (𝔰𝔩4 +, find a set of generators for each one, compute their heights and find a simple Uq (𝔰𝔩4 +-module corresponding to each primitive ideal of Uq (𝔰𝔩4 +.

A parametric family of subalgebras of the Weyl algebra I. Structure and automorphisms

An Ore extension over a polynomial algebra F[x] is either a quantum plane, a quantum Weyl algebra, or an infinite-dimensional unital associative algebra Ah generated by elements x, y, which satisfy yx − xy = h, where h ∈ F[x]. We investigate the family of algebras Ah as h ranges over all the polynomials in F[x]. When h 6= 0, these algebras are subalgebras of the Weyl algebra A1 and can be viewed as differential operators with polynomial coefficients. We give an exact description of the automorphisms of Ah over arbitrary fields F and describe the invariants in Ah under the automorphisms. We determine the center, normal elements, and height one prime ideals of Ah, localizations and Ore sets for Ah, and the Lie ideal [Ah,Ah]. We also show that Ah cannot be realized as a generalized Weyl algebra over F[x], except when h ∈ F. In two sequels to this work, we completely describe the derivations and irreducible modules of Ah over any field.

Double Ore Extensions Versus Iterated Ore Extensions

Motivated by the construction of new examples of Artin–Schelter regular algebras of global dimension four, Zhang and Zhang [6] introduced an algebra extension A P [y 1, y 2; σ, δ, τ] of A, which they called a double Ore extension. This construction seems to be similar to that of a two-step iterated Ore extension over A. The aim of this article is to describe those double Ore extensions which can be presented as iterated Ore extensions of the form A[y 1; σ1, δ1][y 2; σ2, δ2]. We also give partial answers to some questions posed in Zhang and Zhang [6].