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.
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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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Ongong’a, E., Richter, J., Silvestrov, S. (2020). Classification of Low-Dimensional Hom-Lie Algebras. In: Silvestrov, S., Malyarenko, A., Rančić, M. (eds) Algebraic Structures and Applications. SPAS 2017. Springer Proceedings in Mathematics & Statistics, vol 317. Springer, Cham. https://doi.org/10.1007/978-3-030-41850-2_9
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