Images of Lie Polynomials on simple Lie algebras

TL;DR

This paper classifies the possible images of Lie polynomials on simple Chevalley algebras over finite fields with 'very good' characteristic, and constructs explicit Lie polynomials for specific conjugacy classes in (q).

Contribution

It provides a classification of Lie polynomial images on simple Chevalley algebras over finite fields and constructs explicit examples for (q).

Findings

Images are closed under automorphisms and contain zero.

Classification of possible images for simple Chevalley algebras.

Explicit Lie polynomials for (q) conjugacy classes.

Abstract

A Lie polynomial is an element of a free Lie algebra $\mathcal F_k$ on $k$-generators, which defines a Lie map on a given Lie algebra $L$, by substituting $k$-elements of $L$. Similar to word maps on groups and polynomial maps on algebras, one studies here questions analogous to Waring-like problems, the L'vov-Kaplansky conjecture, etc. In this article, we would like to address a problem for Lie algebras parallel to the one Lubotzky solved (Images of word maps in finite simple groups, Glasg. Math. J., 56, no. 2, 465-469, 2014) for finite simple groups. It is easy to verify that the image of a Lie map is (a) closed under automorphism, and (b) contains $0$. In this article, we prove that for a simple Chevalley algebra over a finite field of ``very good'' characteristic, these two properties are enough to classify all possible subsets that can be the image of a Lie polynomial. The next question is to find such Lie polynomials for a given subset satisfying the two properties. Contrary to the results over an algebraically closed field, we find Lie polynomials in the case of Lie algebra $\mathfrak{sl}_2(q)$, for $q$ odd, which give each $\rm{GL}_2(q)$ conjugacy class together with zero as an image.