Basic properties of generalized down–up algebras

Abstract

We introduce a large class of infinite dimensional associative algebras which generalize down-up algebras. Let K be a field and fix f ∈ K[x] and r, s, γ ∈ K. Define L = L(f, r, s, γ ) to be the algebra generated by d, u and h with defining relations:Included in this family are Smith's class of algebras similar to U(sl 2 ), Le Bruyn's conformal sl 2 enveloping algebras and the algebras studied by Rueda. The algebras L have Gelfand-Kirillov dimension 3 and are Noetherian domains if and only if rs = 0. We calculate the global dimension of L and, for rs = 0, classify the simple weight modules for L, including all finite dimensional simple modules. Simple weight modules need not be classical highest weight modules.  2004 Elsevier Inc. All rights reserved.

Automorphisms of Generalized Down-Up Algebras

“…Let f ∈ K[X] be a polynomial and fix scalars r, s, γ ∈ K. The generalized down-up algebra L = L(f, r, s, γ) was defined in [11] as the unital associative K-algebra generated by d, u and h, subject to the relations:…”

“…Several ring-theoretical and homological properties of L were derived by Cassidy and Shelton [11,Secs. 2,3], and in [11,Sec.…”

“…2,3], and in [11,Sec. 4] they classified all simple weight modules of L under the assumption that rs = 0, which is precisely when L is a Noetherian domain.…”

“…Generalized down-up algebras were introduced by Cassidy and Shelton in [11] as a generalization of the down-up algebras A(α, β, γ) of Benkart and Roby [7]. Generalized down-up algebras include all down-up algebras, the algebras similar to the enveloping algebra of sl 2 defined by Smith [25], Le Bruyn's conformal sl 2 enveloping algebras [18] and Rueda's algebras similar to the enveloping algebra of sl 2 [24].…”

“…Generalized down-up algebras include all down-up algebras, the algebras similar to the enveloping algebra of sl 2 defined by Smith [25], Le Bruyn's conformal sl 2 enveloping algebras [18] and Rueda's algebras similar to the enveloping algebra of sl 2 [24]. The reader is encouraged to consult [11] for further details and references.…”