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Hom-Lie algebra structures on semi-simple Lie algebras. (English) Zbl 1144.17005

A hom-Lie algebra can be considered as a deformation of a Lie algebra, which is defined as a pair (L,σ) of a non-associative algebra (L,[·,·]) satisfying skew-symmetry and an algebra homomorphism σ:LL such that


The authors in this paper prove that hom-Lie algebra structures on finite-dimensional simple Lie algebras are all trivial. They also find when a finite-dimensional semi-simple Lie algebra can admit nontrivial hom-Lie algebra structures and obtain the isomorphic classes of nontrivial hom-Lie algebra structures.

Reviewer: Yucai Su (Hefei)
17B20Simple, semisimple, reductive Lie (super)algebras
17B05Structure theory of Lie algebras
[1]Hartwig, J. T.; Larsson, D.; Silvestrov, S. D.: Deformations of Lie algebras using σ-derivations, J. algebra 295, 314-361 (2006) · Zbl 1138.17012 · doi:10.1016/j.jalgebra.2005.07.036
[2]Humphreys, J. E.: Introduction to Lie algebras and representation theory, (1972)
[3]Kac, V. G.: Infinite dimensional Lie algebras, (1990)