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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 \((L,\sigma)\) of a non-associative algebra \((L,[\cdot,\cdot])\) satisfying skew-symmetry and an algebra homomorphism \(\sigma:L\to L\) such that \[ [(id+\sigma)(x),[y,z]]+[(id+\sigma)(y),[z,x]]+[(id+\sigma)(z),[x,y]]=0. \] 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)

MSC:

17B20 Simple, semisimple, reductive (super)algebras
17B05 Structure theory for Lie algebras and superalgebras
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References:

[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
[2] Humphreys, J. E., Introduction to Lie Algebras and Representation Theory (1972), Springer-Verlag: Springer-Verlag New York · Zbl 0254.17004
[3] Kac, V. G., Infinite Dimensional Lie Algebras (1990), Cambridge University Press · Zbl 0574.17002
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