Equations in simple Lie algebras

Tatiana Bandman, Nikolai Gordeev, Boris Kunyavskiǐ, Eugene Plotkin

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12 Scopus citations


We prove that for a given element P(X 1, ..., X d) of the finitely generated free Lie algebra L d, the induced map P:g d→g is dominant for any Chevalley algebra g, provided that K is of characteristic ≠2, and P is not an identity in sl(2,K). We prove that for the Engel monomials [[[X, Y], Y], ..., Y] and for their linear combinations this map is surjective onto the set of non-central elements of g provided that the ground field K is big enough.

Original languageEnglish
Pages (from-to)67-79
Number of pages13
JournalJournal of Algebra
Issue number1
StatePublished - 1 Apr 2012

Bibliographical note

Funding Information:
A substantial part of this work was done during the visits of the first three coauthors to the MPIM (Bonn) in 2010. Bandman, Kunyavski˘ı and Plotkin were supported in part by the Minerva foundation through the Emmy Noether Research Institute. Gordeev was supported in part by RFFI research grants 08-01-00756-a, 10-01-90016-Bel-a, and 11-01-00811-a. The support of these institutions is gratefully appreciated. We also thank M. Agranovsky, D. Calegari, M. Gorelik, A. Joseph, A. Kanel-Belov, and A. Premet for helpful discussions and correspondence.


  • Algebraic group
  • Dominant map
  • Identity
  • Simple Lie algebra


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