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 class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0021869316302897&_mathId=si1.gif&_user=111111111&_pii=S0021869316302897&_rdoc=1&_issn=00218693&md5=3d18b4bb1400b8804c2576c0a65323de" title="Click to view the MathML source">F〈X〉class="mathContainer hidden">class="mathCode">$F〈X〉$ be the free unital associative algebra over F   freely generated by an infinite countable set class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0021869316302897&_mathId=si2.gif&_user=111111111&_pii=S0021869316302897&_rdoc=1&_issn=00218693&md5=8cd5bb5b9b5da02245a4a101237efcb7" title="Click to view the MathML source">X={x₁,x₂,…}class="mathContainer hidden">class="mathCode">$X=\{x_1,x_2,\dots \}$. Define a left-normed commutator class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0021869316302897&_mathId=si3.gif&_user=111111111&_pii=S0021869316302897&_rdoc=1&_issn=00218693&md5=70906eb7779bb87b17f1d2362ff22cdf" title="Click to view the MathML source">[a₁,a₂,…,a_n]class="mathContainer hidden">class="mathCode">$[a_1,a_2,\dots ,a_n]$ recursively by class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0021869316302897&_mathId=si14.gif&_user=111111111&_pii=S0021869316302897&_rdoc=1&_issn=00218693&md5=b6004f600c93bafa650283e688a19ca5" title="Click to view the MathML source">[a₁,a₂]=a₁a₂−a₂a₁class="mathContainer hidden">class="mathCode">$[a_1,a_2]=a_1{a}_2-a_2{a}_1$, class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0021869316302897&_mathId=si15.gif&_user=111111111&_pii=S0021869316302897&_rdoc=1&_issn=00218693&md5=734fb323f0acf8948db1f01290cb7f97" title="Click to view the MathML source">[a₁,…,a_n−1,a_n]=[[a₁,…,a_n−1],a_n]class="mathContainer hidden">class="mathCode">$[a_1,\dots ,a_{n-1},a_n]=[[a_1,\dots ,a_{n-1}],a_n]$ (class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0021869316302897&_mathId=si6.gif&_user=111111111&_pii=S0021869316302897&_rdoc=1&_issn=00218693&md5=3ea9ee74d64ee0f921b3c0832fe2f49c" title="Click to view the MathML source">n≥3class="mathContainer hidden">class="mathCode">$n\ge 3$). For class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0021869316302897&_mathId=si7.gif&_user=111111111&_pii=S0021869316302897&_rdoc=1&_issn=00218693&md5=7358e0e1c6b5bd1b4715c9b15070baf0" title="Click to view the MathML source">n≥2class="mathContainer hidden">class="mathCode">$n\ge 2$, let class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0021869316302897&_mathId=si8.gif&_user=111111111&_pii=S0021869316302897&_rdoc=1&_issn=00218693&md5=5868b09368b699d4c4a7f09fbd1f874a" title="Click to view the MathML source">T⁽ⁿ⁾class="mathContainer hidden">class="mathCode">$T^{(n)}$ be the two-sided ideal in class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0021869316302897&_mathId=si1.gif&_user=111111111&_pii=S0021869316302897&_rdoc=1&_issn=00218693&md5=3d18b4bb1400b8804c2576c0a65323de" title="Click to view the MathML source">F〈X〉class="mathContainer hidden">class="mathCode">$F〈X〉$ generated by all commutators class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0021869316302897&_mathId=si3.gif&_user=111111111&_pii=S0021869316302897&_rdoc=1&_issn=00218693&md5=70906eb7779bb87b17f1d2362ff22cdf" title="Click to view the MathML source">[a₁,a₂,…,a_n]class="mathContainer hidden">class="mathCode">$[a_1,a_2,\dots ,a_n]$ (class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0021869316302897&_mathId=si16.gif&_user=111111111&_pii=S0021869316302897&_rdoc=1&_issn=00218693&md5=1df980dd448862d0519211a0439eb9be" title="Click to view the MathML source">a_i∈F〈X〉class="mathContainer hidden">class="mathCode">$a_i\in F〈X〉$).

Let F   be a field of characteristic 0. In 2008 Etingof, Kim and Ma conjectured that class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0021869316302897&_mathId=si10.gif&_user=111111111&_pii=S0021869316302897&_rdoc=1&_issn=00218693&md5=3f1a15f4d58c4b303432b2c40fc7b17c" title="Click to view the MathML source">T^(m)T⁽ⁿ⁾⊂T^(m+n−1)class="mathContainer hidden">class="mathCode">$T^{(m)}{T}^{(n)}\subset T^{(m+n-1)}$ 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 class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0021869316302897&_mathId=si10.gif&_user=111111111&_pii=S0021869316302897&_rdoc=1&_issn=00218693&md5=3f1a15f4d58c4b303432b2c40fc7b17c" title="Click to view the MathML source">T^(m)T⁽ⁿ⁾⊂T^(m+n−1)class="mathContainer hidden">class="mathCode">$T^{(m)}{T}^{(n)}\subset T^{(m+n-1)}$. The aim of the present note is to confirm the “only if” direction of the conjecture. We prove that if class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0021869316302897&_mathId=si11.gif&_user=111111111&_pii=S0021869316302897&_rdoc=1&_issn=00218693&md5=ac8041e9de28bb7935aae41b032c154e" title="Click to view the MathML source">m=2m^′class="mathContainer hidden">class="mathCode">$m=2m^{\prime }$ and class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0021869316302897&_mathId=si12.gif&_user=111111111&_pii=S0021869316302897&_rdoc=1&_issn=00218693&md5=5ec8264dae7d10791413a383d493235d" title="Click to view the MathML source">n=2n^′class="mathContainer hidden">class="mathCode">$n=2n^{\prime }$ are even then class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0021869316302897&_mathId=si13.gif&_user=111111111&_pii=S0021869316302897&_rdoc=1&_issn=00218693&md5=0d91d7de216b211b6ae713fdd9f3f456" title="Click to view the MathML source">T^(m)T⁽ⁿ⁾⊈T^(m+n−1)class="mathContainer hidden">class="mathCode">$T^{(m)}{T}^{(n)}⊈T^{(m+n-1)}$. Our result is valid over any field F.