Down-up algebras defined over a polynomial base ring \K[t₁, ⋯, t_n]

2014/03/26 by Xin Tang, Tang, Xin
Mathematics · #Advanced Topics in Algebra #Algebraic structures and combinatorial models #Commutative Algebra and Its Applications #FOS: Mathematics #Rings and Algebras (math.RA)

paper · pdf · doi:10.48550/arxiv.1403.6539

openalex publication_date 2014/03/26 · openalex created_date 2016/07/22 · openalex updated_date 2026/07/28

Abstract

In this paper, we study a class of down-up algebras \A defined over a polynomial base ring \K[t₁, ⋯, t_n] and establish several analogous results. We first construct a \K-basis for the algebra \A. As a result, we prove that the Gelfand-Kirillov dimension of \A is n+3 and completely determine the center of \A when char\K=0. Then, we prove that the algebra \A is a noetherian domain if and only if β≠ 0; and \A is Auslander-regular when β≠ 0. We also prove that the global dimension of \A is n+3; and the algebra \A is a prime ring except α=β=ϕ=0. Moreover, we obtain some results on the Krull dimension, isomorphisms, and automorphisms of the algebra \A.

Related

• Classification of factorial generalized down-up algebras
• Down-Up Algebras and Representation Theory
• Gelfand-Kirillov dimension for rings
• Automorphisms of Generalized Down-Up Algebras
• Primitive ideals of noetherian generalized down-up algebras