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Lie algebras with prescribed $ \mathfrak{sl}_3$ decomposition


Authors: Georgia Benkart and Alberto Elduque
Journal: Proc. Amer. Math. Soc. 140 (2012), 2627-2638
MSC (2010): Primary 17B60; Secondary 17A30
DOI: https://doi.org/10.1090/S0002-9939-2011-11120-1
Published electronically: December 9, 2011
MathSciNet review: 2910750
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Abstract: In this work, we consider Lie algebras $ \mathcal {L}$ containing a subalgebra isomorphic to $ \mathfrak{sl}_3$ and such that $ \mathcal {L}$ decomposes as a module for that $ \mathfrak{sl}_3$ subalgebra into copies of the adjoint module, the natural three-dimensional module and its dual, and the trivial one-dimensional module. We determine the multiplication in $ \mathcal {L}$ and establish connections with structurable algebras by exploiting symmetry relative to the symmetric group $ \mathsf {S}_4$.


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Additional Information

Georgia Benkart
Affiliation: Department of Mathematics, University of Wisconsin, Madison, Wisconsin 53706
Email: benkart@math.wisc.edu

Alberto Elduque
Affiliation: Departamento de Matemáticas e Instituto Universitario de Matemáticas y Aplicaciones, Universidad de Zaragoza, 50009 Zaragoza, Spain
Email: elduque@unizar.es

DOI: https://doi.org/10.1090/S0002-9939-2011-11120-1
Keywords: Lie algebra, $\mathfrak{sl}_{3}$ decomposition, structurable algebra
Received by editor(s): January 3, 2001
Received by editor(s) in revised form: March 7, 2011
Published electronically: December 9, 2011
Additional Notes: Part of this work was done during a visit of the first author to the University of Zaragoza, supported by the Spanish Ministerio de Educación y Ciencia and FEDER (MTM 2007-67884-C04-02).
The second author was supported by the Spanish Ministerios de Educación y Ciencia and Ciencia e Innovación and FEDER (MTM 2007-67884-C04-02 and MTM2010-18370-C04-02) and by the Diputación General de Aragón (Grupo de Investigación de Álgebra)
Dedicated: In memory of Hyo Chul Myung
Communicated by: Gail R. Letzter
Article copyright: © Copyright 2011 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.