Abstract
The purpose of this paper is to show that there are Hom-Lie algebra structures on where D is a special type of generalized derivation of and is an algebraically closed field of characteristic zero. We study the representation theory of Hom-Lie algebras within the appropriate category and prove that any finite dimensional representation of a Hom-Lie algebra of the form is completely reducible, analogously to the well-known theorem of Weyl from the classical Lie theory. We apply this result to characterize the non-solvable Lie algebras of this type having an invertible generalized derivation D.
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