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 M₂(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.

KEYWORDS:

• Lie polynomials
• matrices

2010 MATHEMATICS SUBJECT CLASSIFICATION:

• 17B01
• 17B05
• 16R10
• 08A50

Acknowledgments

We would like to thank Aner Shalev for interesting and fruitful discussions regarding this paper.