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Equations in simple Lie algebras

Given an element $P(X_1,...,X_d)$ of the finitely generated free Lie algebra, for any Lie algebra $g$ we can consider the induced polynomial map $P: g^d\to g$. Assuming that $K$ is an arbitrary field of characteristic $\ne 2$, we prove that if $P$ is not an identity in $sl(2,K)$, then this map is dominant for any Chevalley algebra $g$. This result can be viewed as a weak infinitesimal counterpart of Borel's theorem on the dominancy of the word map on connected semisimple algebraic groups. We prove that for the Engel monomials $[[[X,Y],Y],...,Y]$ and, more generally, for their linear combinations, this map is, moreover, surjective onto the set of noncentral elements of $g$ provided that the ground field $K$ is big enough, and show that for monomials of large degree the image of this map contains no nonzero central elements. We also discuss consequences of these results for polynomial maps of associative matrix algebras.