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ORE EXTENSIONS AND POISSON ALGEBRAS

Published online by Cambridge University Press:  13 August 2013

DAVID A. JORDAN
DAVID A. JORDAN*
Affiliation:
School of Mathematics and Statistics, University of Sheffield, Hicks Building, Sheffield S3 7RH, United Kingdom e-mail: d.a.jordan@sheffield.ac.uk
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Abstract

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For a derivation δ of a commutative Noetherian ${\mathbb C}$-algebra A, a homeomorphism is established between the prime spectrum of the Ore extension A[z;δ] and the Poisson prime spectrum of the polynomial algebra A[z] endowed with the Poisson bracket such that {A,A}=0 and {z,a}=δ(a) for all a ∈ A.

Keywords

Primary 17B63Secondary 16S3613N1516W2516S80
Type
Research Article
Information
Glasgow Mathematical Journal , Volume 56 , Issue 2 , May 2014 , pp. 355 - 368
Copyright
Copyright © Glasgow Mathematical Journal Trust 2013 

References

REFERENCES

1.Brown, K. A. and Goodearl, K. R., Lectures on algebraic quantum groups, Advanced Courses in Mathematics, CRM Barcelona (Birkhäuser, Berlin, Germany, 2002).CrossRefGoogle Scholar
2.Coutinho, S. C., d-simple rings and simple $\mathcal{D}$-modules, Math. Proc. Camb. Phil. Soc. 125 (1999), 405–415.Google Scholar
3.Coutinho, S. C., On the differential simplicity of polynomial rings, J. Algebra 264 (2003), 442–468.Google Scholar
4.Coutinho, S. C., On the classification of simple quadratic derivations over the affine plane, J. Algebra 319 (2008), 4249–4274.CrossRefGoogle Scholar
5.Dixmier, J., Enveloping algebras, Graduate Studies in Mathematics, 11 (American Mathematical Society, Providence, RI, 1996).Google Scholar
6.Farkas, D. R., Characterizations of Poisson algebras, Commun. Algebra 23 (1995), 4669–4686.CrossRefGoogle Scholar
7.Goodearl, K. R., Prime ideals in skew polynomial rings and quantized Weyl algebras, J. Algebra 150 (1992), 324–377.Google Scholar
8.Goodearl, K. R., A Dixmier–Moeglin equivalence for Poisson algebras with torus actions, Algebr. Appl. Contemp. Math. 419 (2006), 131–154.Google Scholar
9.Goodearl, K. R., Semi-classical limits of quantized coordinate rings, in Advances in ring theory (Huynh, D. V. and Lopez-Permouth, S., Editors), (Birkhäuser, Basel, Switzerland, 2009) 165–204, arXiv:math.QA/0812.1612v1.Google Scholar
10.Goodearl, K. R. and Warfield, R. B. Jr., Krull dimension of differential operator rings, Proc. London Math. Soc. 45 (3) (1982), 49–70.CrossRefGoogle Scholar
11.Goodearl, K. R. and Warfield, R. B. Jr., Primitivity in differential operator rings, Math. Z. 180 (1982), 503–523.Google Scholar
12.Havran, V. S., Simple derivations of higher degree in two variables, Ukrainian J. Math. 61 (2009), 682–686.Google Scholar
13.Jacobson, N., Structure of rings, rev. ed., (American Mathematical Society, Providence, RI, 1964).Google Scholar
14.Jordan, D. A., Noetherian Ore extensions and Jacobson rings, J. London Math. Soc. 10 (2) (1975), 281–291.CrossRefGoogle Scholar
15.Jordan, D. A., Primitive Ore extensions, Glasgow Math. J. 18 (1977), 93–97.Google Scholar
16.Jordan, D. A., Differentially simple rings with no invertible derivatives, Quart. J. Math. Oxford 32 (1981), 417–424.Google Scholar
17.Jordan, D. A., Finite-dimensional simple Poisson modules, Algebr. Represent. Theory 13 (2010), 79–101.Google Scholar
18.Jordan, D. A. and Oh, S-Q., Poisson brackets and Poisson spectra in polynomial algebras, Contemp. Math. 562 (2012) 169–187.Google Scholar
19.Jordan, D. A. and Sasom, N., Reversible skew Laurent polynomial rings and deformations of Poisson automorphisms, J. Algebra Appl. 8 (2009), 733–757.Google Scholar
20.Kaplansky, I., Commutative rings (Allyn and Bacon, Boston, MA, 1970).Google Scholar
21.Maciejewski, A., Ollagnier, J. Moulin and Nowicki, A., Simple quadratic derivations in two variables, Commun. Algebra 29 (2001), 5095–5113.Google Scholar
22.Nowicki, A., An example of a simple derivation in two variables, Colloq. Math. 113 (2008), 25–31.CrossRefGoogle Scholar
23.Oh, S.-Q., Poisson polynomial rings, Commun. Algebra 34 (2006), 1265–1277.Google Scholar
24.Sigurdsson, G., Differential operator rings whose prime factors have bounded Goldie dimension, Arch. Math. (Basel) 42 (1984), 348–353.Google Scholar