Generalized derivations and Hom-Lie algebra structures on $\mathfrak{sl}_2$

Abstract

The purpose of this paper is to show that there are Hom-Lie algebra structures on sl 2 (F) ⊕ FD, where D is a special type of generalized derivation of sl 2 (F), and F is an algebraically closed field of characteristic zero. It is shown that the generalized derivations D of sl 2 (F) that we study in this work, satisfy the Hom-Lie Jacobi identity for the Lie bracket of sl 2 (F). 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 sl 2 (F)⊕FD, is completely reducible, in analogy to the well known Theorem of Weyl from the classical Lie theory. We apply this result to characterize the non-solvable Lie algebras having an invertible generalized derivation of the type of D. Finally, using root space decomposition techniques we provide an intrinsic proof of the fact that sl 2 (F) is the only simple Lie algebra admitting non-trivial Hom-Lie structures.