The images of Lie polynomials evaluated on matrices |
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| Authors: | Alexei Kanel-Belov Sergey Malev |
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| Institution: | 1. Department of Mathematics, Bar Ilan University, Ramat Gan, Israel;2. Einstein Institute of Mathematics, Hebrew University of Jerusalem, Jerusalem, Israel |
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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 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 sk is not a Lie polynomial, for k>2. |
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| Keywords: | Lie polynomials matrices |
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