2014/07/30 by Leandro Cagliero, Cagliero, Leandro, Fernando Szechtman +1
Mathematics · #17B10 #Advanced Topics in Algebra #Algebraic structures and combinatorial models #FOS: Mathematics #Finite Group Theory Research #Representation Theory (math.RT)
paper · pdf · doi:10.48550/arxiv.1407.8125
openalex publication_date 2014/07/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let mathfrak g be a finite dimensional Lie algebra over a field of\ncharacteristic 0, with solvable radical mathfrak r and nilpotent radical\n mathfrak n=[ mathfrak g, mathfrak r]. Given a finite dimensional\n mathfrak g-module U, its nilpotency series 0\⊂ U( mathfrak\nn1)\⊂\⋯\⊂ U( mathfrak nm)=U is defined so that\nU( mathfrak n1) is the 0-weight space of mathfrak n in U,\nU( mathfrak n2)/U( mathfrak n1) is the 0-weight space of mathfrak\nn in U/U( mathfrak n1), and so on. We say that U is linked if each\nfactor of its nilpotency series is a uniserial mathfrak g/ mathfrak\nn-module, i.e., its mathfrak g/ mathfrak n-submodules form a chain.\nEvery uniserial mathfrak g-module is linked, every linked mathfrak\ng-module is indecomposable with irreducible socle, and both converse fail.\n In this paper we classify all linked mathfrak g-modules when mathfrak\ng=\⟨ x\⟩ ltimes mathfrak a and \ad , x acts\ndiagonalizably on the abelian Lie algebra mathfrak a. Moreover, we\nidentify and classify all uniserial mathfrak g-module amongst them.\n