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.

Abstract. In this paper, we mainly provide a categorical view on the braided structures appearing in the Hom-quantum groups. Let C be a monoidal category on which F is a bimonad, G is a bicomonad, and ϕ is a distributive law, we discuss the necessary and sufficient conditions for C G F (ϕ), the category of mixed bimodules to be monoidal and braided. As applications, we discuss the Hom-type (co)quasitriangular structures, the Hom-Yetter-Drinfeld modules, and the Hom-Long dimodules.2010 Mathematics Subject Classification. 16T25, 15W30.

In this paper, we study the structure of monoidal Hom-Lie algebras in the category H M of H-modules for a triangular Hopf algebra (H, R) and in particular the H-Lie structure of a monoidal Hom-algebra in H M by analogy with that of generalized Lie algebras.An elementary but important property of Lie algebras is that each associative algebra gives rise to a Lie algebra via the commutator bracket. generalized the associativity to twisted associativity and naturally proposed the notion of Hom-associative algebras and they were the first to introduce Hom-coalgebras, Hom-bialgebras, Hom-Hopf algebras and related objects. Furthermore, they obtained that a Hom-associative algebra gives rise to a Hom-Lie algebra via the commutator bracket. Motivated by [2], in which Caenepeel and Goyvaerts studied the Hom-Hopf algebras from a categorical view point, we think whether we can extend the above work by Makhlouf and Silvestrov to the module categories. This becomes our first motivation of writing this paper.Meanwhile, Wang, Kan and Chen [13] studied the structure of the generalized Lie algebras (i.e. the Lie algebras in the module category H M). It is a naive but natural question to ask whether we can obtain same results for the generalized Hom-Lie algebras that are analogous to [13]. This becomes our second motivation of the paper.Let H be any Hopf algebra with a bijective antipode. To give a positive answer to the questions above, we organize this paper as follows.In Sec. 2, we introduce the notion of monoidal Hom-Lie algebras in H M and show that a monoidal Hom-algebra in H M gives rise to a monoidal Hom-Lie algebra in H M by the natural bracket product (see Theorem 2.3). In Sec. 3, we study the structure of the monoidal Hom-algebras and the monoidal Hom-Lie algebras in H M

Let ( , ) be a Hom-Hopf algebra and ( , ) a Hom-coalgebra. In this paper, we first introduce the notions of Hom-crossed coproduct ⋊ and cleft coextension and then discuss the equivalence between them. Furthermore, we discuss the relation between cleft coextension and Hom-module coalgebra with the Hom-Hopf module structure and obtain a Hom-coalgebra factorization for Hom-module coalgebra with the Hom-Hopf module structure.

In this paper, we introduce a representation theory of Hom-Lie conformal superalgebras and discuss the cases of adjoint representations. Furthermore, we develop cohomology theory of Hom-Lie conformal superalgebras and discuss some applications to the study of deformations of regular Hom-Lie conformal superalgebras. Finally, we introduce derivations of multiplicative Hom-Lie conformal superalgebras and study their properties.