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Universal enveloping algebras of Poisson Ore extensions

Authors: Jiafeng Lü, Xingting Wang and Guangbin Zhuang
Journal: Proc. Amer. Math. Soc. 143 (2015), 4633-4645
MSC (2010): Primary 17B63, 17B35, 16S10
Published electronically: March 31, 2015
MathSciNet review: 3391023
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Abstract | References | Similar Articles | Additional Information

Abstract: We prove that the universal enveloping algebra of a Poisson-Ore extension is a length two iterated Ore extension of the original universal enveloping algebra. As a consequence, we observe certain ring-theoretic invariants of the universal enveloping algebras that are preserved under iterated Poisson-Ore extensions. We apply our results to iterated quadratic Poisson algebras arising from semiclassical limits of quantized coordinate rings and a family of graded Poisson algebras of Poisson structures of rank at most two.

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  • [1] Kenneth A. Brown and Iain Gordon, Poisson orders, symplectic reflection algebras and representation theory, J. Reine Angew. Math. 559 (2003), 193-216. MR 1989650 (2004i:16025),
  • [2] Paula A. A. B. Carvalho, Samuel A. Lopes, and Jerzy Matczuk, Double Ore extensions versus iterated Ore extensions, Comm. Algebra 39 (2011), no. 8, 2838-2848. MR 2834133 (2012i:16050),
  • [3] Dragomir Ž. oković, Konstanze Rietsch, and Kaiming Zhao, Normal forms for orthogonal similarity classes of skew-symmetric matrices, J. Algebra 308 (2007), no. 2, 686-703. MR 2295083 (2007m:15018),
  • [4] F. R. Gantmacher, The Theory of Matrices, vol 2 (1989), Chelsea, New York.
  • [5] K. R. Goodearl and S. Launois, The Dixmier-Moeglin equivalence and a Gelfand-Kirillov problem for Poisson polynomial algebras, Bull. Soc. Math. France 139 (2011), no. 1, 1-39 (English, with English and French summaries). MR 2815026 (2012e:17045)
  • [6] Johannes Huebschmann, Poisson cohomology and quantization, J. Reine Angew. Math. 408 (1990), 57-113. MR 1058984 (92e:17027),
  • [7] S. Launois and C. Lecoutre, A quadratic Poisson Gel'fand-Kirillov problem in prime characteristic, preprint, arXiv:1302.2046 (2013).
  • [8] L.-Y. Liu, S.-Q. Wang, and Q.-S. Wu, Twisted Calabi-Yau property of Ore extension, J. Noncommut. Geom., preprint, arXiv:1205.0893 (2012), 19 pp.
  • [9] Jiafeng Lü, Xingting Wang, and Guangbin Zhuang, Universal enveloping algebras of Poisson Hopf algebras, J. Algebra 426 (2015), 92-136. MR 3301903,
  • [10] Sei-Qwon Oh, Poisson enveloping algebras, Comm. Algebra 27 (1999), no. 5, 2181-2186. MR 1683858 (2000b:16050),
  • [11] Sei-Qwon Oh, Poisson polynomial rings, Comm. Algebra 34 (2006), no. 4, 1265-1277. MR 2220812 (2007g:17021),
  • [12] Christopher Phan, The Yoneda algebra of a graded Ore extension, Comm. Algebra 40 (2012), no. 3, 834-844. MR 2899911,
  • [13] M. Towers, Poisson and Hochschild cohomology and the semiclassical limit, preprint, arXiv:1304.6003 (2013).
  • [14] Ualbai Umirbaev, Universal enveloping algebras and universal derivations of Poisson algebras, J. Algebra 354 (2012), 77-94. MR 2879224 (2012m:17034),
  • [15] James J. Zhang and Jun Zhang, Double Ore extensions, J. Pure Appl. Algebra 212 (2008), no. 12, 2668-2690. MR 2452318 (2010h:16066),

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Additional Information

Jiafeng Lü
Affiliation: Department of Mathematics, Zhejiang Normal University, Jinhua, Zhejiang 321004, People’s Republic of China

Xingting Wang
Affiliation: Department of Mathematics, University of Washington, Seattle, Washington 98195
Address at time of publication: Department of Mathematics, University of California, San Diego, CA 92093

Guangbin Zhuang
Affiliation: Department of Mathematics, University of Southern California, Los Angeles, California 90089-2532

Keywords: Poisson algebra, universal enveloping algebra, Ore extension
Received by editor(s): April 4, 2014
Received by editor(s) in revised form: July 30, 2014
Published electronically: March 31, 2015
Additional Notes: The first author was supported by the National Natural Science Foundation of China (No. 11001245, 11271335 and 11101288)
The second author was partially supported by the U. S. National Science Foundation [DMS0855743]
Communicated by: Kailash C. Misra
Article copyright: © Copyright 2015 American Mathematical Society

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