$L$-Modules, $L$-Comodules and Hom-Lie Quasi-Bialgebras

Abstract

In this paper, we discuss $A$-modules and $L$-modules (resp. $L$-comodules) for Hom-Lie algebras (resp. Hom-Lie coalgebras). We show that for a given Hom-associative algebra $A$ (resp. Hom-coassociative coalgebra), the $A$-module (resp. comodule) extends to $L(A)$-module (resp. comodule), where $L(A)$ is the associated Lie algebra (resp. Lie coalgebra), with the same structure map. We also prove that $L$-modules become $L_\alpha$-modules, where $L_\alpha$ is the Hom-Lie algebra obtained from the Lie algebra $L$ by stwisting the Lie bracket. Then we introduce Hom-Lie quasi-bialgebras and prove that a Lie quasi-bialgebra turns to a Hom-Lie quasi-bialgebra by stwisting the Lie quasi-bialgebra structure by an endomorphism. Moreover, we show that an exact Lie quasi-bialgebra extends to an exact Hom-Lie quasi-bialgebra.