×

zbMATH — the first resource for mathematics

Poisson double extensions. (English) Zbl 07184424
J. J. Zhang and J. Zhang [J. Pure Appl. Algebra 212, No. 12, 2668–2690 (2008; Zbl 1157.16009)] defined a very useful concept “double Ore extension” for constructing Artin-Schelter regular algebras. Launois and Lecoutre proved that a class of Ore extensions can be viewed as deformation quantizations of Poisson polynomial extensions. Is a double Ore extension is a deformation quantization of a Poisson algebra? The main aim of this article is to prove that algebras in a class of double Ore extensions are deformation quantizations of certain Poisson algebras called Poisson double extensions
MSC:
17B63 Poisson algebras
16S36 Ordinary and skew polynomial rings and semigroup rings
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Artin, M.; Schelter, W. F., Graded algebras of global dimension 3, Adv Math, 66, 171-216 (1987) · Zbl 0633.16001
[2] Bergman, G. M., The diamond lemma for ring theory, Adv Math, 29, 178-218 (1978) · Zbl 0326.16019
[3] Bitoun, T., The p-support of a holonomic D-module is Lagrangian, for p large enough (2010)
[4] Brown, K. A.; Goodearl, K. R., Lectures on Algebraic Quantum Groups. Advanced Courses in Mathematics (2002), Birkhäuser-Verlag: Basel-Boston-Berlin, Birkhäuser-Verlag
[5] Cavalho, P. A.; Lopes, S. A.; Matczuk, J., Double Ore extensions versus iterated Ore extensions, Comm Algebra, 39, 2838-2848 (2011) · Zbl 1230.16023
[6] Cho, E-H; Oh, S-Q, Semiclassical limits of Ore extensions and a Poisson generalized Weyl algebra, Lett Math Phys, 106, 997-1009 (2016) · Zbl 1356.16019
[7] Dolgushev, V. A., The Van den Bergh duality and the modular symmetry of a Poisson variety, Selecta Math, 14, 199-228 (2009) · Zbl 1172.53054
[8] Goodearl, K. R., A Dixmier-Moeglin equivalence for Poisson algebras with torus actions, Contemp Math, 419, 131-154 (2006) · Zbl 1147.17017
[9] Huebschmann, J., Duality for Lie-Rinehart algebras and the modular class, J Reine Angew Math, 510, 103-159 (1999) · Zbl 1034.53083
[10] Jordan, D. A.; Oh, S-Q, Poisson brackets and Poisson spectra in polynomial algebras, Contemp Math, 562, 169-187 (2012) · Zbl 1273.17027
[11] Jordan, D. A.; Sasom, N., Reversible skew Laurent polynomial rings and deformations of Poisson automorphisms, J Algebra Appl, 8, 733-757 (2009) · Zbl 1188.16022
[12] Kontsevich, M., Deformation quantization of Poisson manifolds I. (1997)
[13] Launois, S.; Lecoutre, C., A quadratic Poisson Gel’fand-Kirillov problem in prime characteristic, Trans Amer Math Soc, 368, 755-785 (2016) · Zbl 1360.17027
[14] Luo, J.; Wang, S. Q.; Wu, Q. S., Twisted Poincaré duality between Poisson homology and Poisson cohomology, J Algebra, 442, 484-505 (2015) · Zbl 1392.17015
[15] Lü, J.; Oh, S-Q; Wang, X., Enveloping algebras of double Poisson-Ore extensions, Comm Algebra, 46, 4891-4904 (2018) · Zbl 1448.17024
[16] Lü, J.; Wang, X.; Zhuang, G., Universal enveloping algebras of Poisson Ore extensions, Proc Amer Math Soc, 143, 4633-4645 (2015) · Zbl 1378.16034
[17] Oh, S-Q, Poisson enveloping algebras, Comm Algebra, 27, 2181-2186 (1999) · Zbl 0936.16020
[18] Oh, S-Q, Poisson polynomial rings, Comm Algebra, 34, 1265-1277 (2006) · Zbl 1135.17012
[19] Oh, S-Q, A natural map from a quantized space onto its semiclassical limit and a multi-parameter Poisson Weyl algebra, Comm Algebra, 45, 60-75 (2017) · Zbl 1405.16037
[20] Rowen, L. H., Graduate Algebra: Commutative View. Graduate Studies in Mathematics (2006), Amer Math Soc: Providence, Amer Math Soc · Zbl 1116.13001
[21] Van den Bergh, M., Noncommutative homology of some three-dimensional quantum spaces, K-Theory, 8, 213-230 (1994) · Zbl 0814.16006
[22] Van den Bergh, M., On involutivity of p-support, Int Math Res Notes IMRN, 15, 6295-6304 (2015) · Zbl 1331.14005
[23] Wang, S. Q., The modular derivations for extensions of Poisson algebras, Front Math China, 12, 209-218 (2017) · Zbl 1431.17015
[24] Zhang, J. J.; Zhang, J., Double Ore extensions, J Pure Appl Algebra, 212, 2668-2690 (2008) · Zbl 1157.16009
[25] Zhu, C.; Van Oystaeyen, F.; Zhang, Y. H., On (co)homology of Frobenius Poisson algebras, J K-Theory: K-Theory Appl Algebra Geom Topol, 14, 371-386 (2014) · Zbl 1309.17012
[26] Zhu, C.; Wang, Y. X., Realization of Poisson enveloping algebra, Front Math China, 13, 999-1011 (2018) · Zbl 06947309
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.