Products of commutators in a Lie nilpotent associative algebra

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• 作者：Galina Deryabina^a ; ^{galina_deryabina@mail.ru" class="auth_mail" title="E-mail the corresponding author} ; Alexei Krasilnikov^b ; ^{alexei@unb.br" class="auth_mail" title="E-mail the corresponding author}
• 关键词：16R10 ; 16R40
• 刊名：Journal of Algebra
• 出版年：2017
• 出版时间：1 January 2017
• 年：2017
• 卷：469
• 期：Complete
• 页码：84-95
• 全文大小：328 K

文摘

Let F   be a field and let b8804c2576c0a65323de" title="Click to view the MathML source">F〈X〉$F〈X〉$ be the free unital associative algebra over F   freely generated by an infinite countable set X={x₁,x₂,…}$X=\{x_1,x_2,\dots \}$. Define a left-normed commutator 906eb7779bb87b17f1d2362ff22cdf" title="Click to view the MathML source">[a₁,a₂,…,a_n]$[a_1,a_2,\dots ,a_n]$ recursively by bafa650283e688a19ca5" title="Click to view the MathML source">[a₁,a₂]=a₁a₂−a₂a₁$[a_1,a_2]=a_1{a}_2-a_2{a}_1$, 90cb7f97" title="Click to view the MathML source">[a₁,…,a_n−1,a_n]=[[a₁,…,a_n−1],a_n]$[a_1,\dots ,a_{n-1},a_n]=[[a_1,\dots ,a_{n-1}],a_n]$ (a9ee74d64ee0f921b3c0832fe2f49c" title="Click to view the MathML source">n≥3$n\ge 3$). For baf0" title="Click to view the MathML source">n≥2$n\ge 2$, let T⁽ⁿ⁾$T^{(n)}$ be the two-sided ideal in b8804c2576c0a65323de" title="Click to view the MathML source">F〈X〉$F〈X〉$ generated by all commutators 906eb7779bb87b17f1d2362ff22cdf" title="Click to view the MathML source">[a₁,a₂,…,a_n]$[a_1,a_2,\dots ,a_n]$ (a_i∈F〈X〉$a_i\in F〈X〉$).

Let F   be a field of characteristic 0. In 2008 Etingof, Kim and Ma conjectured that T^(m)T⁽ⁿ⁾⊂T^(m+n−1)$T^{(m)}{T}^{(n)}\subset T^{(m+n-1)}$ if and only if m or n is odd. In 2010 Bapat and Jordan confirmed the “if” direction of the conjecture: if at least one of the numbers m, n   is odd then T^(m)T⁽ⁿ⁾⊂T^(m+n−1)$T^{(m)}{T}^{(n)}\subset T^{(m+n-1)}$. The aim of the present note is to confirm the “only if” direction of the conjecture. We prove that if m=2m^′$m=2m^{\prime }$ and n=2n^′$n=2n^{\prime }$ are even then T^(m)T⁽ⁿ⁾⊈T^(m+n−1)$T^{(m)}{T}^{(n)}⊈T^{(m+n-1)}$. Our result is valid over any field F.