Shuangjian Guo

Symmetries and the u-condition in Hom-Yetter-Drinfeld categories

Let (H, S, α) be a monoidal Hom-Hopf algebra and [Formula: see text] the Hom-Yetter-Drinfeld category over (H, α). Then in this paper, we first find sufficient and necessary conditions for [Formula: see text] to be symmetric and pseudosymmetric, respectively. Second, we study the u-condition in [Formula: see text] and show that the Hom-Yetter-Drinfeld module (H, adjoint, Δ, α) (resp., (H, m, coadjoint, α)) satisfies the u-condition if and only if S2 = id. Finally, we prove that [Formula: see text] over a triangular (resp., cotriangular) Hom-Hopf algebra contains a rich symmetric subcategory.

Relative Hom-Hopf modules and total integrals

Let (H, α) be a monoidal Hom-Hopf algebra and (A, β) a right (H, α)-Hom-comodule algebra. We first investigate the criterion for the existence of a total integral of (A, β) in the setting of monoidal Hom-Hopf algebras. Also, we prove that there exists a total integral ϕ : (H, α) → (A, β) if and only if any representation of the pair (H, A) is injective in a functorial way, as a corepresentation of (H, α), which generalizes Doi’s result. Finally, we define a total quantum integral γ : H → Hom(H, A) and prove the following affineness criterion: if there exists a total quantum integral γ and the canonical map ψ : A⊗BA → A ⊗ H, a⊗Bb ↦ β−1(a) b[0] ⊗ α(b[1]) is surjective, then the induction functor A⊗B−:ℋ˜(ℳk)B→ℋ˜(ℳk)AH is an equivalence of categories.

The construction and deformation of BiHom-Novikov algebras

Abstract BiHom-Novikov algebra is a generalized Hom-Novikov algebra endowed with two commuting multiplicative linear maps. The main purpose of this paper is to show that two classes of BiHom-Novikov algebras can be constructed from BiHom-commutative algebras together with derivations and BiHom-Novikov algebras with Rota–Baxter operators, respectively. We show that quadratic BiHom-Novikov algebras are associative algebras and the sub-adjacent BiHom-Lie algebras of BiHom-Novikov algebras are 2-step nilpotent. Moreover, we develop the 1-parameter formal deformation theory of BiHom-Novikov algebras.

Symmetric pairs and pseudosymmetries in Hom-Yetter–Drinfeld categories

In this paper, we study symmetric pairs and pseudosymmetries in the Hom-Yetter–Drinfeld category HH𝕐𝔻 over a Hom-Hopf algebra (H,S,β). We first show that the category HH𝕐𝔻 over a (co)triangular Hom-Hopf algebra H contains a rich symmetric subcategory. Also we prove that the (co)commutativity and trivial property of H are determined by some symmetric pairs of objects in HH𝕐𝔻. Moreover, we find a sufficient and necessary condition for HH𝕐𝔻 to be pseudosymmetric.

Partial representation of partial twisted smash products

In this paper we first give the sufficient conditions under which a partial twisted smash product algebra and the usual tensor product coalgebra become a bialgebra. Furthermore, we introduce the notion of partial representation of partial twisted smash products and explore its relationship with partial actions of Hopf algebras. Finally, we give the conditions for the partial twisted smash products to be Frobenius.

Total integrals of Doi Hom-Hopf modules

Let (H,α) be a monoidal Hom-Hopf algebra, (A,β) a right (H,α)-Hom-comodule algebra and (C,γ) a right (H,α)-Hom-module coalgebra. We first investigate the criterion for the existence of a total integral of (A,β) in the setting of monoidal Hom-Hopf algebras. Also, we prove that there exists a total integral Υ : C →Hom(C,A) if and only if any representation of the pair (H,A,C) is injective in a functorial way, which generalizes Menini and Militaru’s result. Finally, we extend to the category of (H,A,C)-Doi Hom-Hopf modules a result of Doi on projectivity of every relative (A,H)-Hopf module as an (A,β)-module.