Down-up algebras and ambiskew polynomial rings.

*(English)*Zbl 0958.16030The down-up algebras \(A=A(\alpha,\beta,\gamma)\) introduced by G. Benkart and T. Roby [J. Algebra 209, No. 1, 305-344 (1998; Zbl 0922.17006)] are analyzed using the technology of iterated skew polynomial rings. It is known that \(A\) is Noetherian precisely when \(\beta\neq 0\) [E. Kirkman, I. M. Musson, and D. S. Passman, Proc. Am. Math. Soc. 127, No. 11, 3161-3167 (1999; Zbl 0940.16012)]. In this case, the author shows that \(A\) can be written as an iterated skew polynomial ring of the form \(\mathbb{C}[t][x;\sigma^{-1}][y;\sigma,\delta]\) where \(\sigma\) is an automorphism of \(\mathbb{C}[t]\), extended to \(\mathbb{C}[t][x;\sigma^{-1}]\) so that \(x\) is a \(\sigma\)-eigenvector. Previous work of the author [Math. Z. 213, No. 3, 353-371 (1993; Zbl 0797.16037)] and the author and I. E. Wells [Proc. Edinb. Math. Soc., II. Ser. 39, No. 3, 461-472 (1996; Zbl 0864.16027)] leads to (a) criteria for \(A\) to be primitive; (b) classification of the finite dimensional simple \(A\)-modules; (c) criteria for the semisimplicity of the finite dimensional \(A\)-modules; and (d) in many cases, determination of the height \(1\) prime ideals of \(A\). The results under (a), (b) and (c) overlap with work of E. Kirkman and J. Kuzmanovich [Commun. Algebra 28, No. 6, 2983-2997 (2000; Zbl 0965.16001)] and P. A. A. B. Carvalho and I. M. Musson [J. Algebra 228, No. 1, 286-310 (2000; Zbl 0965.16002)].

In the non-Noetherian case (\(\beta=0\)), the author shows that \(A\) can again be represented as an iterated skew polynomial ring in two indeterminates, where now one iteration requires right-hand coefficients while the other requires left-hand coefficients. The author’s techniques [e.g., J. Pure Appl. Algebra 98, No. 1, 45-55 (1995; Zbl 0829.16017)] again result in a classification of the finite dimensional simple \(A\)-modules, as well as a computation of the prime spectrum of \(A\).

In the non-Noetherian case (\(\beta=0\)), the author shows that \(A\) can again be represented as an iterated skew polynomial ring in two indeterminates, where now one iteration requires right-hand coefficients while the other requires left-hand coefficients. The author’s techniques [e.g., J. Pure Appl. Algebra 98, No. 1, 45-55 (1995; Zbl 0829.16017)] again result in a classification of the finite dimensional simple \(A\)-modules, as well as a computation of the prime spectrum of \(A\).

Reviewer: K.R.Goodearl (Santa Barbara)

##### MSC:

16S36 | Ordinary and skew polynomial rings and semigroup rings |

16P40 | Noetherian rings and modules (associative rings and algebras) |

16D60 | Simple and semisimple modules, primitive rings and ideals in associative algebras |

16D25 | Ideals in associative algebras |

##### Keywords:

down-up algebras; iterated skew polynomial rings; ambiskew polynomial rings; simple modules; height 1 prime ideals; prime spectra
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DOI

##### References:

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