Noetherian down-up algebras

Abstract. Down-up algebras A = A(; ; ) were introduced by G. Benkart and T. Roby to better understand the structure of certain posets. In this paper, we prove that 6 = 0 is equivalent to A being right (or left) Noetherian, and also to A being a domain. Furthermore, when this occurs, we show that A is Auslander-regular and has global dimension 3. Motivated by the study of posets, G. Benkart and T. Roby introduced certain down-up algebras in [BR], see also [B]. Specically, let K be a eld, x parameters ; ; 2 K and let A = A(; ; ) be the K-algebra with generators d and u, and relation