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arXiv · 1506.06792

The images of Lie polynomials evaluated on matrices

Alexei Kanel-Belov·Sergey Malev·Louis Rowen
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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 $k>2.$

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Alexei Kanel-Belov, Sergey Malev, Louis Rowen. 2015-12-03. The images of Lie polynomials evaluated on matrices. https://doi.org/10.1080/00927872.2017.1282959

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