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Down-up algebras and ambiskew polynomial rings. (English) Zbl 0958.16030
The 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 $$A$$.

##### 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
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