Abstract
We consider algebras of quantum differential operators, for appropriate bicharacters on a polynomial algebra in one indeterminate and for the coordinate algebra of quantum n-space for n ≥ 3. In the former case a set of generators for the quantum differential operators was identified in work by the first author and T. C. McCune but it was not known whether the algebra is Noetherian. We answer this question affirmatively, setting it in a more general context involving the behaviour Noetherian condition under localization at the powers of a single element. In the latter case we determine the algebra of quantum differential operators as a skew group algebra of the group \(\mathbb {Z}^{n}\) over a quantized Weyl algebra. It follows from this description that this algebra is a simple right and left Noetherian domain.
Similar content being viewed by others
References
Artamonov, V.A.: Generalized derivations of the quantum plane (Russian) Sovrem. Matematicas Prilozh. No. 13, Algebra (2004), 40-52; translation in Journal Mathematics Science (N. Y.) 131 (2005), no. 5, 5904-5918
Artamonov, V.A.: Actions of pointed Hopf algebras on quantum torus. Ann. Univ. Ferrara Sez. VII (N.S.) 51, 29–60 (2005)
Artamonov, V.A.: Quantum polynomials Advances in algebra and combinatorics, vol. 19-34. World Science Publications, Hackensack, NJ (2008)
Alev, J., Chamarie, M.: Derivations et automorphismes de quelques algebres quantiques. Comm. Algebra 20(6), 1787–1802 (1992)
Goodearl, K.R.: Prime ideals in skew polynomial rings and quantized Weyl algebras. J. Algebra 150(2), 324–377 (1992)
Goodearl, K.R., Warfield, R.B.: An Introduction to Noncommutative Noetherian Rings. 2nd. London Mathematics Society Student Texts 61, Cambridge (2004)
Iyer, U.N.: Volichenko algebras as algebras of differential operators. J. Nonlinear Math. Phys. 13(1), 34–49 (2006)
Iyer, U.N., McCune, T.C.: Quantum differential operators on 𝕜[x]. Int. J. Math. 13(4), 395–413 (2002)
Iyer, U.N., McCune, T.C.: Quantum differential operators on the quantum plane. J. Algebra 260(2), 577–591 (2003)
Jordan, D.A.: A simple localization of the quantized Weyl algebra. J. Algebra 174(2), 267–281 (1995)
Jordan, D.A.: The graded algebra generated by two Eulerian derivatives. Algebr. Represent. Theory 4, 249–275 (2001)
Jordan, D.A., Wells, I.E.: Simple ambiskew polynomial rings. J. Algebra 382(2), 46–70 (2013)
Lunts, V., Rosenberg, A.: Differential operators on noncommutative rings. Sel. Math. (N.S) 3, 335–359 (1997)
Maltsiniotis, G.: Groupes quantique et structures différentielles. C.R. Acad. Sci. Paris Sér. I Math. 311, 831–834 (1990)
McConnell, J.C., Robson, J.C.: with the cooperation of L. W. Small, Noncommutative Noetherian Rings. In: Also, Graduate Studies in Mathematics, Volume 30, American Mathematical Society, p. 2001. Wiley, Chichester, Providence RI (1987)
McConnell, J.C., Pettit, J.J.: Crossed products and multiplicative analogues of Weyl algebras. J. Lond. Math. Soc. 38, 47–55 (1988)
Rogalski, D.: Generic noncommutative surfaces. Adv. Math. 184(2), 289–341 (2004)
Author information
Authors and Affiliations
Corresponding author
Additional information
Presented by Paul Smith.
Rights and permissions
About this article
Cite this article
Iyer, U.N., Jordan, D.A. Noetherian Algebras of Quantum Differential Operators. Algebr Represent Theor 18, 1593–1622 (2015). https://doi.org/10.1007/s10468-015-9553-8
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10468-015-9553-8