On the transitivity of Lie ideals and a characterization of perfect Lie algebras

2024/05/31 by Nikolaos Panagiotis Souris, Souris, Nikolaos Panagiotis · 1 citation
Mathematics · #17B05 (Primary) #17B40 #22E60 (Secondary) #Advanced Topics in Algebra #Algebraic structures and combinatorial models #FOS: Mathematics #Finite Group Theory Research #Rings and Algebras (math.RA)

paper · pdf · doi:10.48550/arxiv.2405.20893

openalex publication_date 2024/05/31 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We explore general intrinsic and extrinsic conditions that allow the transitivity of the relation of being a Lie ideal, in the sense that if a Lie algebra \mathfrakh is a subideal of a Lie algebra \mathfrakg (i.e. there exist Lie subalgebras \mathfrakl₀,\mathfrakl₁,…,\mathfrakl_n of \mathfrakg with \mathfrakh=\mathfrakl₀\unlhd \mathfrakl₁ \unlhd⋯ \unlhd \mathfrakl_n=\mathfrakg), then \mathfrakh is an ideal of \mathfrakg. We also prove that perfect Lie algebras of arbitrary dimension and over any field are intrinsically characterized by transitivity of this type; In particular, we show that a Lie algebra \mathfrakh is perfect (i.e. \mathfrakh=[\mathfrakh, \mathfrakh]) if and only if for any Lie algebra \mathfrakg such that \mathfrakh is a subideal of \mathfrakg, it follows that \mathfrakh is an ideal of \mathfrakg.

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