Down–Up Algebras and Their Representation Theory

Carvalho, Paula A. A. B.; Musson, Ian M.

doi:10.1006/jabr.1999.8263

Cited by 40 publications

(39 citation statements)

References 13 publications

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“…As has been observed in [9], when β = 0 and γ = 0 down up algebras are AS regular of type S 1 (cf [10]). Consequently, we have the following.…”

Section: Regularity and Dimension Resultssupporting

confidence: 58%

Homogenized Down-Up Algebras

Cassidy¹

2003

Communications in Algebra

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Abstract. This paper studies two homogenizations of the down-up algebras introduced in [1]. We show that in all cases the homogenizing variable is not a zero-divisor, and that when the parameter β is non-zero, the homogenized down-up algebra is a Noetherian domain and a maximal order, and also Artin-Schelter regular, Auslander regular, and Cohen-Macaulay. We show that all homogenized down-up algebras have global dimension 4 and Gelfand-Kirillov dimension 4, and with one exception all homogenized down-up algebras are prime rings. We also exhibit a basis for homogenized down-up algebras and provide a necessary condition for a Noetherian homogenized down-up algebra to be a Hopf algebra.

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“…As has been observed in [9], when β = 0 and γ = 0 down up algebras are AS regular of type S 1 (cf [10]). Consequently, we have the following.…”

Section: Regularity and Dimension Resultssupporting

confidence: 58%

Homogenized Down-Up Algebras

Cassidy¹

2003

Communications in Algebra

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“…We remark that the isomorphism problem for Noetherian down-up algebras has already been solved in [10].…”

Section: Automorphisms Of Down-up Algebrasmentioning

confidence: 97%

“…Again, T i ≥ 0 by (10). Moreover, there is 0 ≤ i ≤ l such that a i = 0 and n i = 0, by the condition that q is not a multiple of h. It follows from (10) that ǫ divides n 0 , say n 0 = ǫm.…”

Section: The Normal Elements Of Lmentioning

confidence: 99%

Automorphisms of Generalized Down-Up Algebras

Carvalho

Lopes

2009

Communications in Algebra

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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.

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“…This presentation is similar to the presentation of A 1 − β β 1 as an iterated skew polynomial ring in [9].…”

Section: Case Bmentioning

confidence: 60%

Down–Up Algebras and Ambiskew Polynomial Rings

Jordan

2000

Journal of Algebra

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We show that the down-up algebras of G. Benkart (1998, in "Recent Progress in Algebra," Contemporary Mathematics Vol. 224, Am. Math. Soc., Providence) and G. Benkart and T. Roby (1998, J. Algebra 209, 305-344) lie in a certain class of iterated skew polynomial rings, called ambiskew polynomial rings, in two indeterminates x and y over a commutative ring B. In such rings, commutation of the indeterminates with elements of B involve the same endomorphism σ of B, but from different sides, that is, yb = σ b y and bx = xσ b , and, for some scalar p, yx − pxy ∈ B. In previous studies of ambiskew polynomial rings, σ was required to be an automorphism but, in order to cover all down-up algebras, this requirement must be dropped. The Noetherian down-up algebras are those where σ is an automorphism and, in this case, we apply existing results on ambiskew polynomial rings to determine the finite-dimensional simple modules and the prime ideals. We adapt the methods underlying these results so as to apply to the non-Noetherian down-up algebras for which they reveal a surprisingly rich structure.

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Down–Up Algebras and Their Representation Theory

Cited by 40 publications

References 13 publications

Homogenized Down-Up Algebras

Homogenized Down-Up Algebras

Automorphisms of Generalized Down-Up Algebras

Down–Up Algebras and Ambiskew Polynomial Rings

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