Equations in simple Lie algebras

Tatiana Bandman; Nikolai Gordeev; Boris Kunyavskiǐ; Eugene Plotkin

doi:10.1016/j.jalgebra.2012.01.012

Equations in simple Lie algebras

Tatiana Bandman, Nikolai Gordeev, Boris Kunyavskiǐ, Eugene Plotkin

Department of Mathematics

Herzen Russian State Pedagogical University

Research output: Contribution to journal › Article › peer-review

13 Scopus citations

Abstract

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 language	English
Pages (from-to)	67-79
Number of pages	13
Journal	Journal of Algebra
Volume	355
Issue number	1
DOIs	https://doi.org/10.1016/j.jalgebra.2012.01.012
State	Published - 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.

Funding

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.

Funders	Funder number
Emmy Noether Research Institute for Mathematics
Russian Foundation for Fundamental Investigations	08-01-00756-a, 11-01-00811-a, 10-01-90016-Bel-a
Minerva Foundation

Keywords

Algebraic group
Dominant map
Identity
Simple Lie algebra

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10.1016/j.jalgebra.2012.01.012

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title = "Equations in simple Lie algebras",

abstract = "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.",

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author = "Tatiana Bandman and Nikolai Gordeev and Boris Kunyavskiǐ and Eugene Plotkin",

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.",

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AU - Kunyavskiǐ, Boris

AU - Plotkin, Eugene

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N2 - 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.

AB - 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.

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Equations in simple Lie algebras

Abstract

Bibliographical note

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Keywords

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