Let $F_n$ be the free Lie algebra over $F$ of rank $n$ generated by $y_1,...y_n$, and let $f\inF_n'$ be a multilinear Lie polynomial contained in the commutator ideal of $F_n$. In this paper, we determine the image $Im(f)=\{f(w_1,\ldots,w_n)|w_i\in L\}\subseteq L$ for Lie algebras $L$ of dimension $\leq 3$, and of the Lie algebra of dimension 4 stated in a paper of Baker dating back to 1901. In all the cases studied, the L'vov-Kaplansky Conjecture has a positive answer.
The image of Lie polynomials on real Lie algebras of dimension up to 3
Centrone, Lucio
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2024-01-01
Abstract
Let $F_n$ be the free Lie algebra over $F$ of rank $n$ generated by $y_1,...y_n$, and let $f\inF_n'$ be a multilinear Lie polynomial contained in the commutator ideal of $F_n$. In this paper, we determine the image $Im(f)=\{f(w_1,\ldots,w_n)|w_i\in L\}\subseteq L$ for Lie algebras $L$ of dimension $\leq 3$, and of the Lie algebra of dimension 4 stated in a paper of Baker dating back to 1901. In all the cases studied, the L'vov-Kaplansky Conjecture has a positive answer.File in questo prodotto:
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