Hom-RS-Lie algebras and related Hom-Leibniz algebras

In F.-E. Abid and M. Boucetta, Complete description of invariant, associative pseudo-Euclidean metrics on left Leibniz algebras via quadratic Lie algebras, J. Algebra 638 (2024) 358-395; S. Benayadi and S. Hidri, Leibniz algebras with invariant bilinear forms and related Lie algebras, Commun. Algebra 44(8) (2016) 3538-3556 (resp. S. Benayadi and F. Mhamdi, Invariant bilinear forms on Leibniz superalgebras, J. Algebra Appl. 21(05) (2022) 2250098) the authors investigated Leibniz algebras (resp. superalgebras) provided with associative, left-invariant, and right-invariant bilinear forms. In this paper, we generalize these types of invariance to Hom-Leibniz algebras, introducing the concepts of α-invariance, α-left-invariance, and α-right-invariance for a Hom-Leibniz algebra (ℒ, ·, α). Focusing on α-R-quadratic Hom-Leibniz algebras, which are (left or right) Hom-Leibniz algebras equipped with symmetric, non-degenerate, and α-right-invariant bilinear forms,we highlight the connections between these Hom-algebras and some other Hom-algebraic structures. More precisely, every α-R-quadratic regular symmetric Hom-Leibniz algebra gives rise to a new type of Hom-algebra, which we call Hom-RS-Lie algebra. We investigate Hom-RS-Lie algebras and provide some interesting information on their structure. Specifically, we characterize firstly α-R-quadratic regular symmetric Hom-Leibniz algebras using the double extension process, which allows us to describe inductively their associated Hom-RS-Lie algebras. Finally, several non-trivial examples of Hom-RS-Lie algebras are included.