The Quasitriangular Structure For Hom-T-Smash Product Hom-Hopf Algebra And Hom-Cotwisted Coalgebra

Deformation theory of algebras is now one of the important branch of algebras. In recent years, Hom-algebras as a class of deformation algebras have attracted the attention of many scholars, Hom-(co) algebras is actually a generalization of (co) algebra that replace (co)associativity with the Hom-(co)associativity. i.e., a(a)(bc)=(ab)a(c), (a(a,1)(?)a21(?)a22=a11(?)a12(?)a(a2)).In this paper, the quasitriangular structure of Hom-Hopf algebras and the twisted structure of Hom-coalgebras are studied. The main contents are as follows:(1) Firstly, we give the concepts of dual compatibility Hom-U-Hopf algebra, the skew dual Hom-V-Hopf algebra, the weaken quasitriangular Hom-Hopf algebra. The quasi-triangular structures of Hom-Hopf algebras are discussed. The necessary and sufficient conditions for (B (?)T H, R) to be a quasitriangular Hom-Hopf algebra are given in terms of properties of their components and the quasitriangular structure of B B(?)T H can be decomposed as R=P(1)U(1)(?)Q(1)V(1)(?)P(2)V(2)(?)Q(2)U(2).(2) Hom-types of the cotwistors are discussed. We first give the notion of Hom-cotwistor, then Hom-comultiplication is twisted by using the Hom-cotwistor. We obtain a new Hom-coalgebra structure Dw=(D, W o Δ,ε,β), called Hom-cotwisted coalgebra. In the final, we show the structure theorem of Hom-twisted bialgebra. That is if W:H â†' H is a Hom-algebra map and T:Hwâ†' Hw a Hom-coalgebra map, then Hom-twisted algebra (HT,μo T,1H,γ) and Hom-cotwisted coalgebra (Hw,W o Δ,ε,γ) form a Hom-twisted bialgebra (HWT,μo T, 1H, W oΔ,ε,γ) (Theorem 3.4.5.)...