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Ideals in the Enveloping Algebra of the Positive Witt Algebra

  • Alexey V. Petukhov
  • Susan J. SierraEmail author
Open Access
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Abstract

Let W+ be the positive Witt algebra, which has a \(\mathcal {C}\)-basis \(\{e_{n}: n \in \mathcal {Z}_{\geq 1}\}\), with Lie bracket [ei,ej] = (ji)ei+j. We study the two-sided ideal structure of the universal enveloping algebra U(W+) of W+. We show that if I is a (two-sided) ideal of U(W+) generated by quadratic expressions in the ei, then U(W+)/I has finite Gelfand-Kirillov dimension, and that such ideals satisfy the ascending chain condition. We conjecture that analogous facts hold for arbitrary ideals of U(W+), and verify a version of these conjectures for radical Poisson ideals of the symmetric algebra S(W+).

Keywords

Witt algebra Positive Witt algebra Poisson algebra Poisson Gelfand-Kirillov dimension Ascending chain condition 

Mathematics Subject Classification (2010)

Primary: 16S30 17B63 17B68 Secondary 16P70 16P90 17B65 17B70 

Notes

Acknowledgments

The first author was supported by Leverhulme Trust Grant RPG-2013-293 and RFBR grant 16-01-00818. The second author was supported by EPSRC grant EP/M008460/1.

We would like to thank Jacques Alev, Tom Lenagan, Omar Leon Sanchez, Paul Smith and Toby Stafford for helpful discussions. We would particularly like to thank Ioan Stanciu, whose computer experiments, done as part of his MMath dissertation at the University of Edinburgh, gave us experimental evidence for Conjecture 1.2.

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Copyright information

© The Author(s) 2019

Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors and Affiliations

  1. 1.Jacobs University BremenBremenGermany
  2. 2.Institute for Information Transmission problemsMoscowRussia
  3. 3.School of MathematicsThe University of EdinburghEdinburghUK

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