Equations in simple Lie algebras

We prove that for a given element P(X1,…,Xd) of the finitely generated free Lie algebra Ld, the induced map P:gd→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 thei...

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Bibliographic Details
Published in:Journal of algebra 2012-04, Vol.355 (1), p.67-79
Main Authors: Bandman, Tatiana, Gordeev, Nikolai, Kunyavskiĭ, Boris, Plotkin, Eugene
Format: Article
Language:English
Subjects:
Algebraic group
Dominant map
Identity
Simple Lie algebra
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Summary:We prove that for a given element P(X1,…,Xd) of the finitely generated free Lie algebra Ld, the induced map P:gd→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.
ISSN:0021-8693
1090-266X
DOI:10.1016/j.jalgebra.2012.01.012