General Hom-Lie algebra. (Q2801827)

A Hom-algebra \(A\) is an algebra in which the associative law is twisted by an algebra map \(f\) from \(A\) to \(A\), i.e., \(A\) satisfies \(f(a)bc=abf(c)\) for \(a,b,c\) in \(A\). Similarly one has Hom-coalgebras, Hom-bialgebras, Hom-Hopf algebras, Hom-modules, module Hom-algebras (for the last two, see [\textit{D. Yau}, ``Module Hom-algebras'', \url{arXiv:0812.4695}]), quasitriangular Hom-Hopf algebras \(H\) and the Hom-module category for \(H\) (for these last two, see [\textit{H. Li} and \textit{T. Ma} ``On Hom-Yetter-Drinfeld category'' preprint (2014)]). \textit{H. Li} and \textit{T. Ma} introduced the Yetter-Drinfeld category \(_HM\) of Hom-\(H\)-modules [Colloq. Math. 137, No. 1, 43-65 (2014; Zbl 1306.16031)]. A Hom-Lie algebra is an algebra with a bracket product satisfying a twisted version of the Jacobi identity.NEWLINENEWLINE In the paper under review, the authors define a general Hom-Lie algebra to be a Lie algebra in the category \(_HM\) for \(H\) a quasitriangular Hom-Hopf algebra. They give two examples. The main theorem of the paper is that if \((H,f,R)\) is a triangular Hom-Hopf algebra (\(f\colon H\to H\) the twisting) and \((A,g)\) is an \((H,f)\)-module Hom-algebra, then \((A,[\;,\;],g,R)\) is a general Hom-Lie algebra with suitably defined bracket product \([\;,\;]\). This extends some previous work which used the usual category of left \(H\)-modules [\textit{S. Wang} et al., Algebra Colloq. 9, No. 2, 143-154 (2002; Zbl 1007.16032); and \textit{L. Dong} et al., J. Algebra Appl. 13, No. 5, Article ID 1350149 (2014; Zbl 1304.16033)].