On the images of multilinear maps of matrices over finite-dimensional division algebras

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• 作者：Cailan Li^a ; ^{thecailanli@berkeley.edu" class="auth_mail" title="E-mail the corresponding author} ; Man Cheung Tsui^b ; ^{mantsui@live.com" class="auth_mail" title="E-mail the corresponding author}
• 关键词：15A54 ; 16S50 ; 16K20
• 刊名：Linear Algebra and its Applications
• 出版年：2016
• 出版时间：15 March 2016
• 年：2016
• 卷：493
• 期：Complete
• 页码：399-410
• 全文大小：321 K

文摘

Let R   be a central simple algebra finite-dimensional over its center class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S002437951500717X&_mathId=si1.gif&_user=111111111&_pii=S002437951500717X&_rdoc=1&_issn=00243795&md5=14ce0fef2ee6ac216dc050cb7f155524" title="Click to view the MathML source">Fclass="mathContainer hidden">class="mathCode">$\mathbb{F}$ of characteristic 0. We will show that every element of reduced trace 0 in R   can be expressed as class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S002437951500717X&_mathId=si2.gif&_user=111111111&_pii=S002437951500717X&_rdoc=1&_issn=00243795&md5=abf0bdfe15694fbbd3dad152b07c9243" title="Click to view the MathML source">[a,[c,b]]+λ[c,[a,b]]class="mathContainer hidden">class="mathCode">$[a,[c,b]]+\lambda [c,[a,b]]$ for some class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S002437951500717X&_mathId=si3.gif&_user=111111111&_pii=S002437951500717X&_rdoc=1&_issn=00243795&md5=716bf034d03ab8276433284178df9c6c" title="Click to view the MathML source">a,b,c∈Rclass="mathContainer hidden">class="mathCode">$a,b,c\in R$ where class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S002437951500717X&_mathId=si4.gif&_user=111111111&_pii=S002437951500717X&_rdoc=1&_issn=00243795&md5=86714c7f61ea06078a81cfd890d18acd" title="Click to view the MathML source">λ≠0,−1class="mathContainer hidden">class="mathCode">$\lambda \ne 0,-1$. In addition, let class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S002437951500717X&_mathId=si18.gif&_user=111111111&_pii=S002437951500717X&_rdoc=1&_issn=00243795&md5=7239748a47c4df5182a2e98c5775f081" title="Click to view the MathML source">Dclass="mathContainer hidden">class="mathCode">$\mathcal{D}$ be a division algebra satisfying the conditions above. We will also show that the set of values of any nonzero multilinear polynomial of degree at most three, with coefficients from the center class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S002437951500717X&_mathId=si1.gif&_user=111111111&_pii=S002437951500717X&_rdoc=1&_issn=00243795&md5=14ce0fef2ee6ac216dc050cb7f155524" title="Click to view the MathML source">Fclass="mathContainer hidden">class="mathCode">$\mathbb{F}$ of class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S002437951500717X&_mathId=si18.gif&_user=111111111&_pii=S002437951500717X&_rdoc=1&_issn=00243795&md5=7239748a47c4df5182a2e98c5775f081" title="Click to view the MathML source">Dclass="mathContainer hidden">class="mathCode">$\mathcal{D}$, evaluated on class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S002437951500717X&_mathId=si19.gif&_user=111111111&_pii=S002437951500717X&_rdoc=1&_issn=00243795&md5=8d0da536ab5932538014220575f80170" title="Click to view the MathML source">M_k(D)class="mathContainer hidden">class="mathCode">$M_k(\mathcal{D})$, class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S002437951500717X&_mathId=si39.gif&_user=111111111&_pii=S002437951500717X&_rdoc=1&_issn=00243795&md5=d0a1cfd28b9efef0ab0fb639da515f1a" title="Click to view the MathML source">k≥2class="mathContainer hidden">class="mathCode">$k\ge 2$, contains all matrices of reduced trace 0.