The generalised nilradical of a Lie algebra

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• 作者：David A. Towers ^{d.towers@lancaster.ac.uk" class="auth_mail" title="E-mail the corresponding author}
• 关键词：17B05 ; 17B20 ; 17B30 ; 17B50
• 刊名：Journal of Algebra
• 出版年：2017
• 出版时间：15 January 2017
• 年：2017
• 卷：470
• 期：Complete
• 页码：197-218
• 全文大小：414 K

文摘

A solvable Lie algebra L has the property that its nilradical N contains its own centraliser. This is interesting because it gives a representation of L as a subalgebra of the derivation algebra of its nilradical with kernel equal to the centre of N. Here we consider several possible generalisations of the nilradical for which this property holds in any Lie algebra. Our main result states that for every Lie algebra L  , class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S002186931630309X&_mathId=si1.gif&_user=111111111&_pii=S002186931630309X&_rdoc=1&_issn=00218693&md5=4d4d00629bb442706d8034179e40b522" title="Click to view the MathML source">L/Z(N)class="mathContainer hidden">class="mathCode">$L/Z(N)$, where class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S002186931630309X&_mathId=si2.gif&_user=111111111&_pii=S002186931630309X&_rdoc=1&_issn=00218693&md5=d756d8511980b31fd1497368da04e896" title="Click to view the MathML source">Z(N)class="mathContainer hidden">class="mathCode">$Z(N)$ is the centre of the nilradical of L  , is isomorphic to a subalgebra of class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S002186931630309X&_mathId=si3.gif&_user=111111111&_pii=S002186931630309X&_rdoc=1&_issn=00218693&md5=1e7b9740c023459362bb98ebfccb465c" title="Click to view the MathML source">Der(N^⁎)class="mathContainer hidden">class="mathCode">$\mathrm{Der}(N^{⁎})$ where class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S002186931630309X&_mathId=si4.gif&_user=111111111&_pii=S002186931630309X&_rdoc=1&_issn=00218693&md5=43ebb4bfe864353336d4f3e97065594f" title="Click to view the MathML source">N^⁎class="mathContainer hidden">class="mathCode">$N^{⁎}$ is an ideal of L   such that class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S002186931630309X&_mathId=si429.gif&_user=111111111&_pii=S002186931630309X&_rdoc=1&_issn=00218693&md5=26dad47508fb80f38a7d9af675879eb3" title="Click to view the MathML source">N^⁎/Nclass="mathContainer hidden">class="mathCode">$N^{⁎}/N$ is the socle of a semisimple Lie algebra.