General Hom–Lie algebra

Abstract

We introduce the notion of a general Hom–Lie algebra and show that any Hom-algebra in the category [Formula: see text], where [Formula: see text] is a triangular Hom–Hopf algebra, can give rise to a general Hom–Lie algebra, which generalizes the main result in [11] to the Hom-case.

Another approach to Hom-Lie bialgebras via Manin triples

“…With this generalization of the Lie algebra, some q-deformations of the Witt and the Virasoro algebras have the structure of a Hom-Lie algebra [8]. Due to their close relationship with discrete and deformed vector fields and differential calculus [8,9,10], Hom-Lie algebras have been studied in broad areas [1,2,3,4,5,11,12,14,15,16,17,18,20].…”

“…11. A homomorphism of Hom-Lie bialgebras f : (g, ∆ g ) −→ (k, ∆ k ) is a homomorphism of Hom-Lie algebras such that…”

Matching Hom-Setting of Rota-Baxter Algebras, Dendriform Algebras, and Pre-Lie Algebras

“…It is shown that a Homassociative algebra gives rise to a Hom-Lie algebra using the commutator. Since then, various Hom-analogues of some classical algebraic structures have been introduced and studied intensively, such as Hom-coalgebras, Hom-bialgebras and Hom-Hopf algebras [24,25], Hom-groups [26,27], Hom-Hopf modules [28], Hom-Lie superalgebras [29,30], generalize Hom-Lie algebras [31], and Hom-Poisson algebras [32].…”

On the algebraic variety of Hom-Lie algebras

“…They have called this new class of algebras the class of Hom-Lie algebras. This notion, introduced in [3], since made the object of numerous studies and was also generalized (see [4]). We denote by SAlg n the set of the n-dimensional skew-symmetric algebras (A, µ) over an algebraically closed field K of characteristic 0 whose multiplication µ is skew-symmetric and by HLie n the subset of Hom-Lie algebras.…”