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Hom-Lie Algebras with a Set Grading


In this paper we study the regular Hom-Lie algebra \({\mathscr{L}}\) graded by an arbitrary set \(\mathcal {S}\) (set grading). We show that \({\mathscr{L}}\) decomposes as \({\mathscr{L}}=\mathcal {U}\oplus {\sum }_{[\lambda ]\in ({\varLambda }_{\mathcal {S}}\setminus \{0\})/\sim }{\mathscr{L}}_{[\lambda ]}\), where \(\mathcal {U}\) is a linear complement of \({\sum }_{[\lambda ]\in ({\varLambda }_{\mathcal {S}}\setminus \{0\})/\sim }{\mathscr{L}}_{0,[\lambda ]}\) in \({\mathscr{L}}_{0}\) and any \({\mathscr{L}}_{[\lambda ]}\) a well-described graded ideals of \({\mathscr{L}}\), satisfying \([{\mathscr{L}}_{[\lambda ]}, {\mathscr{L}}_{[\mu ]}]=0\) if [λ]≠[μ]. Under certain conditions, the simplicity of \({\mathscr{L}}\) is characterized and it is shown that the above decomposition is actually the direct sum of the family of its minimal graded ideals, each one being a simple set-graded regular Hom-Lie algebras.

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Correspondence to Valiollah Khalili.

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Khalili, V. Hom-Lie Algebras with a Set Grading. Vietnam J. Math. 50, 111–124 (2022).

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  • Hom-Lie algebras
  • Set-grade Lie algebras
  • Structure theory

Mathematics Subject Classification (2010)

  • 17B40
  • 17B55
  • 17B75