The images of Lie polynomials evaluated on matrices

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 $k>2.$

References in corpus (2)

• Word equations in simple groups and polynomial equations in simple algebras
• Surjectivity of certain word maps on PSL(2,C) and SL(2,C)

Cited by in corpus (7)

• Evaluations of Noncommutative Polynomials on Algebras: Methods and Problems, and the L'vov-Kaplansky Conjecture
• Instances of the Kaplansky-Lvov multilinear conjecture for polynomials of degree three
• The images of multilinear and semihomogeneous polynomials on the algebra of octonions
• The Mesyan Conjecture: a restatement and a correction
• On the combinatorics of commutators of Lie algebras
• Image of Lie polynomial of degree $2$ evaluated on Nilpotent Lie algebra
• The images of multilinear non-associative polynomials evaluated on a rock-paper-scissors algebra with unit over an arbitrary field