×

zbMATH — the first resource for mathematics

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
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Bavula, V.V., Generalized Weyl algebras and their representations, Saint |St. Petersburg math. J., 4, 71-92, (1993) · Zbl 0807.16027
[2] Bavula, V.V., Description of bilateral ideals in a class of noncommutative rings I, Ukrain. mat. zh., 45, 209-220, (1993) · Zbl 0809.16001
[3] Bavula, V.V., Generalized Weyl algebras, kernel and tensor-simple algebras, their simple modules, Canad. math. soc. conf. proc., 14, 83-107, (1993) · Zbl 0806.17023
[4] Bavula, V.V., Global dimension of generalized Weyl algebras, Canad. math. soc. confe. proce., 18, 81-107, (1996) · Zbl 0857.16025
[5] V. V. Bavula, and, D. A. Jordan, Isomorphism Problems and Groups of Automorphisms for Generalized Weyl Algebras, Trans. Amer. Math. Soc, to appear. · Zbl 0961.16016
[6] Bavula, V.V.; Van Oystaeyen, F., Krull dimension of generalized Weyl algebras and iterated skew polynomial rings. commutative coefficients, J. algebra, 208, 1-35, (1998) · Zbl 0923.16022
[7] Benkart, G., Down – up algebras and Witten’s deformations of the universal enveloping algebra of \(s\)\(l\)_2, Recent progress in algebra, Contemporary mathematics, 224, (1998), Am. Math. Soc Providence
[8] Benkart, G.; Roby, T., Down – up algebras, J. algebra, 209, 305-344, (1998) · Zbl 0922.17006
[9] Carvalho, P.A.A.B.; Musson, I.M., Down – up algebras and their representation theory, J. algebra, 228, 286-310, (2000) · Zbl 0965.16002
[10] Cohn, P.M., Free rings and their relations, (1985), Academic Press London · Zbl 0659.16001
[11] Dumas, F.; Jordan, D.A., The 2×2 quantum matrix Weyl algebra, Commun. algebra, 24, 1409-1434, (1996) · Zbl 0851.16025
[12] Goodearl, K.R., Prime ideals in skew polynomial rings and quantized Weyl algebras, J. algebra, 150, 324-377, (1992) · Zbl 0779.16010
[13] Goodearl, K.R.; Warfield, R.B., An introduction to noncommutative Noetherian rings, (1989), Cambridge Univ. Press Cambridge · Zbl 0679.16001
[14] Jacobson, N., Structure of rings, Am. math. soc. colloq. publi., XXXVII, (1968), Am. Math. Soc Providence
[15] Jordan, D.A., Krull and global dimension of certain iterated skew polynomial rings, Abelian groups and noncommutative rings, a collection of papers in memory of robert B. warfield, jr., Contemporary mathematics, 130, (1992), Am. Math. Soc Providence, p. 201-213 · Zbl 0779.16011
[16] Jordan, D.A., Iterated skew polynomial rings and quantum groups, J. algebra, 174, 267-281, (1993)
[17] Jordan, D.A., Height one prime ideals of certain iterated skew polynomial rings, Math. proc. Cambridge philos. soc., 114, 407-425, (1993) · Zbl 0804.16028
[18] Jordan, D.A., Primitivity in skew Laurent polynomial rings and related rings, Math. Z., 213, 353-371, (1993) · Zbl 0797.16037
[19] Jordan, D.A., Finite-dimensional simple modules over certain iterated skew polynomial rings, J. pure appl. algebra, 98, 45-55, (1995) · Zbl 0829.16017
[20] Jordan, D.A., A simple localization of the quantized Weyl algebra, J. algebra, 174, 267-281, (1995) · Zbl 0833.16025
[21] Jordan, D.A.; Wells, I.E., Invariants for automorphisms of certain iterated skew polynomial rings, Proc. Edinburgh math. soc., 39, 461-472, (1996) · Zbl 0864.16027
[22] Kassel, C., Quantum groups, (1995), Springer-Verlag New York · Zbl 0808.17003
[23] E. Kirkman, and, J. Kuzmanovich, Primitivity of Noetherian Down-Up Algebras, preprint. · Zbl 0965.16001
[24] E. Kirkman, and, J. Kuzmanovich, Non-Noetherian Down-Up Algebras, preprint. · Zbl 0967.16013
[25] Kirkman, E.; Musson, I.M.; Passman, D.S., Noetherian down – up algebras, Proc. amer. math. soc., 127, 3161-3167, (1999) · Zbl 0940.16012
[26] R. S. Kulkarni, Down-Up Algebras and Their Representations, to appear. · Zbl 1036.16022
[27] Kurosh, A.G., The theory of groups, (1955), Chelsea New York · Zbl 0068.26104
[28] Lane, D.R., Fixed points of affine Cremona transformations of the plane over an algebraically closed field, Amer. J. math., 97, 707-732, (1975) · Zbl 0319.17001
[29] McConnell, J.C.; Robson, J.C., Noncommutative Noetherian rings, (1987), Wiley Chichester · Zbl 0644.16008
[30] Smith, M.K., Eigenvectors of automorphisms of polynomial rings in two variables, Houston J. math., 10, 559-573, (1984) · Zbl 0573.13010
[31] Smith, S.P., Quantum groups. an introduction and survey for ring theorists, Non-commutative rings, MSRI pub., 24, (1992), Springer Berlin Heidelberg New York, p. 131-178 · Zbl 0744.16023
[32] Terwilliger, P., The incidence algebra of a uniform poset, (), 193-212 · Zbl 0737.05032
[33] I. E. Wells, Generalized Weyl Algebras, Ph.D. thesis, University of Sheffield, 1995.
[34] Witten, E., Gauge theories, vertex models, and quantum groups, Nuclear phys. B, 330, 285-346, (1990)
[35] Woronowicz, S.L., Twisted SU(2)-group. an example of a non-commutative differential calculus, Publ. res. inst math. sci., 23, 117-181, (1987) · Zbl 0676.46050
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.