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RESTRICTED LIE ALGEBRAS WITH MAXIMAL 0-PIM

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Abstract

In this paper it is shown that the projective cover of the trivial irreducible module of a finite-dimensional solvable restricted Lie algebra is induced from the one dimensional trivial module of a maximal torus. As a consequence, the number of the isomorphism classes of irreducible modules with a fixed p-character for a finite-dimensional solvable restricted Lie algebra L is bounded above by p MT(L), where MT(L) denotes the maximal dimension of a torus in L. Finally, it is proved that in characteristic p > 3 the projective cover of the trivial irreducible L-module is induced from the one-dimensional trivial module of a torus of maximal dimension, only if L is solvable.

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Authors and Affiliations

  1. Department of Mathematics and Statistics, University of South Alabama, Mobile, AL, 36688-0002, USA

    JӦRG FELDVOSS

  2. Dipartimento di Matematica e Fisica “Ennio De Giorgi”, Università del Salento, Via Provinciale Lecce-Arnesano, I-73100, Lecce, Italy

    SALVATORE SICILIANO

  3. Dipartimento di Matematica e Applicazioni, Università degli Studi di, Milano-Bicocca Via Roberto Cozzi 55, I-20125, Milano, Italy

    THOMAS WEIGEL

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  1. JӦRG FELDVOSS
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Correspondence to JӦRG FELDVOSS.

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Dedicated to Otto H. Kegel on the occasion of his eighty-first birthday

2010 Mathematics Subject Classification. Primary: 17B50. Secondary: 17B05, 17B10, 17B30. Key words and phrases: Restricted Lie algebra, p-character, reduced universal enveloping algebra, projective cover, projective indecomposable module, induced module, maximal 0-PIM, torus, solvable Lie algebra, number of irreducible modules.

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FELDVOSS, J., SICILIANO, S. & WEIGEL, T. RESTRICTED LIE ALGEBRAS WITH MAXIMAL 0-PIM. Transformation Groups 21, 377–398 (2016). https://doi.org/10.1007/s00031-015-9362-5

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