The images of Lie polynomials evaluated on matrices

What is it about?

In this paper the authors study the problem of determining the image of a Lie polynomial evaluated on a given Lie algebra. The authors' main theorem claims that for any algebraically closed field $K$ of characteristic not equal to 2, the image of any Lie polynomial evaluated on ${\rm sl}_2(K)$ is either ${\rm sl}_2(K)$, or $\{0\}$, or the set of trace zero non-nilpotent matrices, together with 0. Examples are provided in the paper. They also verify that the standard polynomial $s_k=\sum_{\pi\in S_k}{\rm sgn}(\pi)x_{\pi(1)}\ldots x_{\pi(k)}$ is not a Lie polynomial for $k>2$. Reviewed by Aleksandr Panov

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