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On some classes of \({\mathbb {Z}}\)-graded Lie algebras

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Abstract

We study finite dimensional almost- and quasi-effective prolongations of nilpotent \({\mathbb {Z}}\)-graded Lie algebras, especially focusing on those having a decomposable reductive structural subalgebra. Our assumptions generalize effectiveness and algebraicity and are appropriate to obtain Levi–Malčev and Levi–Chevalley decompositions and precisions on the heigth and other properties of the prolongations in a very natural way. In a last section we consider the semisimple case and discuss some examples in which the structural algebras are central extensions of orthogonal Lie algebras and their degree \((-\,1)\) components arise from spin representations.

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Authors and Affiliations

  1. Dipartimento di Scienze Matematiche, Fisiche e Informatiche, Università di Parma, Parco Area delle Scienze 53/A (Campus), 43124, Parma, Italy

    Stefano Marini & Costantino Medori

  2. Dipartimento di Matematica, II Università di Roma “Tor Vergata”, Via della Ricerca Scientifica, 00133, Rome, Italy

    Mauro Nacinovich

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  1. Stefano Marini
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  2. Costantino Medori
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  3. Mauro Nacinovich
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Correspondence to Costantino Medori.

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Communicated by Vicente Cortés.

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Marini, S., Medori, C. & Nacinovich, M. On some classes of \({\mathbb {Z}}\)-graded Lie algebras. Abh. Math. Semin. Univ. Hambg. 90, 45–71 (2020). https://doi.org/10.1007/s12188-020-00217-9

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