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Classification of Low-Dimensional Hom-Lie Algebras

  • Elvice Ongong’aEmail author
  • Johan Richter
  • Sergei Silvestrov
Conference paper
  • 35 Downloads
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 317)

Abstract

We derive conditions for an arbitrary n-dimensional algebra to be a Hom-Lie algebra, in the form of a system of polynomial equations, containing both structure constants of the skew-symmetric bilinear map and constants describing the twisting linear endomorphism. The equations are linear in the constants representing the endomorphism and non-linear in the structure constants. When the algebra is 3 or 4-dimensional we describe the space of possible endomorphisms with minimum dimension. For the 3-dimensional case we give families of 3-dimensional Hom-Lie algebras arising from a general nilpotent linear endomorphism constructed upto isomorphism together with non-isomorphic canonical representatives for all the families in that case. We further give a list of 4-dimensional Hom-Lie algebras arising from a general nilpotent linear endomorphisms.

Keywords

Hom-Lie algebras Classification Isomorphism Structure constants Nilpotent linear endomorphisms 

MSC 2010 Classification

17A30 

Notes

Acknowledgements

Elvice Ongong’a is grateful to the International Science Program, Uppsala University for the support in the framework of the Eastern Africa Universities Mathematics Programme (EAUMP) and to the research environment in Mathematics and Applied Mathematics MAM, the Division of Applied Mathematics of the School of Education, Culture and Communication at Mälardalen University for hospitality and creating excellent conditions for research, research education and cooperation.

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Copyright information

© Springer Nature Switzerland AG 2020

Authors and Affiliations

  • Elvice Ongong’a
    • 1
    • 2
    Email author
  • Johan Richter
    • 3
  • Sergei Silvestrov
    • 2
  1. 1.School of MathematicsUniversity of NairobiNairobiKenya
  2. 2.Division of Applied Mathematics, School of Education, Culture and CommunicationMälardalen UniversityVästeråsSweden
  3. 3.Department of Mathematics and Natural SciencesBlekinge Institute of TechnologyKarlskronaSweden

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