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Abstract <p> We study the existence of an integrable complex structure on a real finite-dimensional naturally graded Lie algebra that is narrow in the sense of Zelmanov and Shalev. Every such Lie algebra is generated by two elements. As the most elementary examples, one can consider naturally graded filiform Lie algebras. We show that almost all such nilpotent Lie algebras admit no integrable complex structures. The only exceptions for which the question under study has a positive answer are the even-dimensional quotient Lie algebras <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11501_2025_8537_Article_IEq1.gif" Format="GIF" Height="20" Rendition="HTML" Resolution="72" Type="Linedraw" Width="41" /> </InlineMediaObject> <EquationSource Format="TEX">\(\mathfrak n_1^+(s)\)</EquationSource> </InlineEquation> obtained by factoring the infinite-dimensional subalgebra <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11501_2025_8537_Article_IEq2.gif" Format="GIF" Height="20" Rendition="HTML" Resolution="72" Type="Linedraw" Width="20" /> </InlineMediaObject> <EquationSource Format="TEX">\(\mathfrak n_1^+\)</EquationSource> </InlineEquation> of the loop algebra <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11501_2025_8537_Article_IEq3.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="62" /> </InlineMediaObject> <EquationSource Format="TEX">\(\mathcal L(\mathfrak{so}(3))\)</EquationSource> </InlineEquation> of the real simple Lie algebra <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11501_2025_8537_Article_IEq4.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="38" /> </InlineMediaObject> <EquationSource Format="TEX">\(\mathfrak{so}(3)\)</EquationSource> </InlineEquation> by the ideals <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11501_2025_8537_Article_IEq5.gif" Format="GIF" Height="20" Rendition="HTML" Resolution="72" Type="Linedraw" Width="53" /> </InlineMediaObject> <EquationSource Format="TEX">\((\mathfrak n_1^+)^{s+1}\)</EquationSource> </InlineEquation> of the lower central series. </p>

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Narrow Lie Algebras and Integrable Complex Structures

  • 期刊论文

  • 发表日期：

    2025年6月1日

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  • Dmitry Millionshchikov

摘要

Abstract

We study the existence of an integrable complex structure on a real finite-dimensional naturally graded Lie algebra that is narrow in the sense of Zelmanov and Shalev. Every such Lie algebra is generated by two elements. As the most elementary examples, one can consider naturally graded filiform Lie algebras. We show that almost all such nilpotent Lie algebras admit no integrable complex structures. The only exceptions for which the question under study has a positive answer are the even-dimensional quotient Lie algebras \(\mathfrak n_1^+(s)\) obtained by factoring the infinite-dimensional subalgebra \(\mathfrak n_1^+\) of the loop algebra \(\mathcal L(\mathfrak{so}(3))\) of the real simple Lie algebra \(\mathfrak{so}(3)\) by the ideals \((\mathfrak n_1^+)^{s+1}\) of the lower central series.