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Gradings on Modules Over Lie Algebras of E Types

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Abstract

For any grading by an abelian group G on the exceptional simple Lie algebra \(\mathcal {L}\) of type E 6 or E 7 over an algebraically closed field of characteristic zero, we compute the graded Brauer invariants of simple finite-dimensional modules, thus completing the computation of these invariants for simple finite-dimensional Lie algebras. This yields the classification of finite-dimensional G-graded simple \(\mathcal {L}\)-modules, as well as necessary and sufficient conditions for a finite-dimensional \(\mathcal {L}\)-module to admit a G-grading compatible with the given G-grading on \(\mathcal {L}\).

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Acknowledgements

Cristina Draper was supported by the SpanishMinisterio de Economía y Competitividad—Fondo Europeo de Desarrollo Regional (FEDER) MTM2013-41208-P and MTM2016-76327-C3-1-P, and by the Junta de Andalucía grants FQM-336 and FQM-7156, with FEDER funds. Orcid code: 0000-0002-2998-7473.

Alberto Elduque was supported by the Spanish Ministerio de Economía y Competitividad—Fondo Europeo de Desarrollo Regional (FEDER) MTM2013-45588-C3-2-P, and by the Diputaci´on General de Aragón—Fondo Social Europeo (Grupo de Investigación de Álgebra). Orcid code: 0000-0002-6497-2162.

Mikhail Kochetov was supported by Discovery Grant 341792-2013 of the Natural Sciences and Engineering Research Council (NSERC) of Canada.

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Authors and Affiliations

  1. Departamento de Matemática Aplicada, Escuela de Ingenierías Industriales, Universidad de Málaga, Ampliación Campus de Teatinos, 29071, Málaga, Spain

    Cristina Draper

  2. Departamento de Matemáticas e Instituto Universitario de Matemáticas y Aplicaciones, Universidad de Zaragoza, 50009, Zaragoza, Spain

    Alberto Elduque

  3. Department of Mathematics and Statistics, Memorial University of Newfoundland, St. John’s, NL, A1C5S7, Canada

    Mikhail Kochetov

Authors
  1. Cristina Draper
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  2. Alberto Elduque
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  3. Mikhail Kochetov
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Corresponding author

Correspondence to Alberto Elduque.

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Presented by Henning Krause.

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Draper, C., Elduque, A. & Kochetov, M. Gradings on Modules Over Lie Algebras of E Types. Algebr Represent Theor 20, 1085–1107 (2017). https://doi.org/10.1007/s10468-017-9675-2

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  • DOI: https://doi.org/10.1007/s10468-017-9675-2

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