The Dynkin theorem for multilinear Lie elements (Q2856388)

Let \(F\) be the associative algebra over a field of characteristic 0 freely generated by \(a_1,\dots,a_n\) and \(F_n\) the subspace of \(F\) spanned by \(a_\sigma(1)\dots a_\sigma(n)\), where \(\sigma\) runs over the symmetric group \(S_n\). For each \(\ell\in\{1,\dots,n\}\), the map \(p_\ell\) is defined on \(F_n\) by the rule: NEWLINE\[NEWLINEp_\ell(\sum_{\sigma\in S_n}\lambda_\sigma a_{\sigma(1)}\dots a_{\sigma(n)})=\sum_{\sigma\in S_n,\;\sigma(n)=\ell}\lambda_\sigma[ a_{\sigma(1)},[ a_{\sigma(1)},\dots,[a_{\sigma(n-1)},a_\ell]\dots],NEWLINE\]NEWLINE where \([x,y]\) stands for the commutator \(xy-yx\). The author proves that \(X\in F_n\) is a Lie elements (i.e., belongs to the Lie algebra generated by \(a_1,\dots,a_n\)) if and only if \(p_\ell X=X\) for each \(\ell\in\{1,\dots,n\}\) (Theorem~1.1). This is a multilinear version of the well-known Dynkin--Specht--Wever theorem (that deals with arbitrary homogeneous elements). As applications, the author exhibits concrete Lie representations for several multilinear polynomials known to be Lee elements, including the \(q\)-Solomon element and the multi-parameter Klyachko element. The latter result was earlier obtained by the author in [J. Algebr. Comb. 33, No. 4, 531--542 (2011; Zbl 1264.17004)].