Ideals in the Enveloping Algebra of the Positive Witt Algebra

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


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+).


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 



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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© The Author(s) 2019

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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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