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<p>Generalized semidirect sums of Lie algebras and their modules are introduced, which are not necessarily (non)-Abelian extensions and may be applied to construct Lie algebras from modules. Some properties of generalized semidirect sums are described. In particular, it is shown that finite dimensional non-solvable Lie algebras can be realized as generalized semidirect sums. The complete classification up to isomorphism of all generalized semidirect sums of <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10587_2025_9624_Article_IEq1.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="20" /> </InlineMediaObject> <EquationSource Format="TEX">\(\mathfrak{sl}_2\)</EquationSource> <EquationSource Format="MATHML"><math display="block"> <msub> <mrow> <mi mathvariant="fraktur">s</mi> <mi mathvariant="fraktur">l</mi> </mrow> <mn>2</mn> </msub> </math></EquationSource> </InlineEquation> and its finite-dimensional irreducible modules is given.</p>

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Generalized semidirect sums of Lie algebras and their modules

  • 期刊论文

  • 发表日期：

    2025年1月25日

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  • Rui Lu,
  • Youjun Tan

摘要

Generalized semidirect sums of Lie algebras and their modules are introduced, which are not necessarily (non)-Abelian extensions and may be applied to construct Lie algebras from modules. Some properties of generalized semidirect sums are described. In particular, it is shown that finite dimensional non-solvable Lie algebras can be realized as generalized semidirect sums. The complete classification up to isomorphism of all generalized semidirect sums of \(\mathfrak{sl}_2\) s l 2 and its finite-dimensional irreducible modules is given.