Cobraided smash product Hom-Hopf algebras

Abstract

Abstract. Let (A, α) and (B, β) be two Hom-Hopf algebras. In this paper, we construct a class of new Hom-Hopf algebras: R-smash product (A♮ R B, α ⊗ β). Moreover, necessary and sufficient conditions for (A♮ R B, α ⊗ β) to be a cobraided Hom-Hopf algebra are given.

A construction of the Hom-Yetter–Drinfeld category

“…(HCM C1) is satisfied if and only if ∆ C is a morphism of H-comodules for suitable H-comodule structures on C ⊗ C and C, respectively. Definition 2.8 (see [11]) Let (H, β) be a Hom-bialgebra and (C, α) a Hom-coalgebra. If (C, ⊲, α) is a left (H, β)-Hom-module and for all h ∈ H and c ∈ A,…”

“…Definition 2.10 (see [11]) Let (H, β) be a Hom-bialgebra and (A, ⊲, α) an (H, β)module Hom-algebra. Then (A♮H, α ⊗ β) (A♮H = A ⊗ H as a linear space) with the multiplication…”

“…Since ∆ A ♮ ⋄ H = ∆ A⋄H is a morphism of Hom-algebras, for all a, b ∈ A and h, g ∈ H, we have (2) Theorem 3.3 is different from the one defined by Liu and Shen in [9], because the Hom-smash product there is based on the concept of module Hom-algebra in [3] and ours is based on the Yau's in [24,29]. Corollary 3.4 (see [11]) Let (A, α), (H, β) be two Hom-bialgebras, and (A, ⊲, α) an (H, β)-module Hom-algebra. Then the smash product Hom-algebra (A♮H, α ⊗ β) endowed with the tensor product Hom-coalgebra structure becomes a Hom-bialgebra if and only if (A, ⊲, α) is an (H, β)-module Hom-coalgebra and…”

“…Let (H, β) be a Hom-bialgebra, (A, α) a left (H, β)-module Hom-algebra and a left (H, β)-comodule Hom-coalgebra. In [11], the smash product Homalgebra (A♮H, α ⊗ β) was constructed. In Section 3, we first define smash coproduct Hom-coalgebra (A ⋄ H, α ⊗ β) (see Proposition 3.1), then derive necessary and sufficient conditions for (A♮H, α⊗β) and (A⋄H, α⊗β) to be a Hom-bialgebra, which is called Radford biproduct Hom-bialgebra and denoted by (A ♮ ⋄ H, α ⊗ β) (see Theorem 3.3, 3.6).…”

The Factorization for a Class of Hom-Coalgebras

“…Recall from [15], let ( , ) be a Hom-Hopf algebra and ( , ) a Hom-coalgebra, and if ( , ⋅, ) is a left ( , )-Hom-module, for all ∈ , ℎ ∈ , the following conditions hold:…”