The images of Lie polynomials evaluated on $2\times 2$ matrices over an algebraically closed field

Alexei Kanel-Belov, Sergey Malev, Louis Rowen

Published 2015-06-22Version 1

Let $f$ be an arbitrary polynomial in several non commutative variables. Kaplansky asked about the possible images of $f$. In this note we let $f$ be a Lie polynomial in several non-commuting variables with constant term $0$ and coefficients in an algebraically closed field $K$. We describe all possible images of $f$ and provide an example of $f$ whose image is the set of trace zero matrices without nilpotent non zero matrices. We provide an arithmetic criterion for this case.