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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 $\left(L,\sigma \right)$ of a non-associative algebra $\left(L,\left[·,·\right]\right)$ satisfying skew-symmetry and an algebra homomorphism $\sigma :L\to L$ such that

$\left[\left(id+\sigma \right)\left(x\right),\left[y,z\right]\right]+\left[\left(id+\sigma \right)\left(y\right),\left[z,x\right]\right]+\left[\left(id+\sigma \right)\left(z\right),\left[x,y\right]\right]=0·$

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 Lie (super)algebras 17B05 Structure theory of Lie algebras
Hom-Lie algebras
##### References:
 [1] Hartwig, J. T.; Larsson, D.; Silvestrov, S. D.: Deformations of Lie algebras using $\sigma$-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)