The images of Lie polynomials evaluated on matrices

TL;DR

This paper classifies the possible images of Lie polynomials evaluated on 2x2 matrices, providing explicit descriptions, criteria, and examples, and shows that certain standard polynomials are not Lie polynomials.

Contribution

It characterizes all images of Lie polynomials on 2x2 matrices and demonstrates that standard polynomials are not Lie polynomials for degrees greater than two.

Findings

All possible images of Lie polynomials on M_2(K) are described.

An example of a Lie polynomial with image equal to non-nilpotent trace zero matrices plus zero is provided.

The standard polynomial s_k is not a Lie polynomial for k > 2.

Abstract

Kaplansky asked about the possible images of a polynomial $f$ in several noncommuting variables. In this paper we consider the case of $f$ a Lie polynomial. We describe all the possible images of $f$ in $M_2(K)$ and provide an example of $f$ whose image is the set of non-nilpotent trace zero matrices, together with 0. We provide an arithmetic criterion for this case. We also show that the standard polynomial $s_k$ is not a Lie polynomial, for $k>2.$