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<p>By introducing a module structure on the tensor product of a symmetric module of a Leibniz algebra, we obtain a construction of Leibniz algebras which generalizes the usual semidirect sums. The resulting Leibniz algebras, called the generalized semidirect sums, are given by a pair of module homomorphisms. Some properties of generalized semidirect sums are described, which are applied to split Leibniz algebras and their symmetric modules. Generalized semidirect sums from the complex 3-dimensional simple Lie algebra and its finite-dimensional irreducible modules are classified up to isomorphism of Leibniz algebras.</p>

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Leibniz algebras arising from symmetric modules and module homomorphisms

  • 期刊论文

  • 发表日期：

    2026年6月19日

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  • Miting Deng,
  • Rui Lu,
  • Youjun Tan

摘要

By introducing a module structure on the tensor product of a symmetric module of a Leibniz algebra, we obtain a construction of Leibniz algebras which generalizes the usual semidirect sums. The resulting Leibniz algebras, called the generalized semidirect sums, are given by a pair of module homomorphisms. Some properties of generalized semidirect sums are described, which are applied to split Leibniz algebras and their symmetric modules. Generalized semidirect sums from the complex 3-dimensional simple Lie algebra and its finite-dimensional irreducible modules are classified up to isomorphism of Leibniz algebras.