Monoidal Hom-Hopf algebras

Hom-structures (Lie algebras, algebras, coalgebras, Hopf algebras) have been investigated in the literature recently. We study Hom-structures from the point of view of monoidal categories; in particular, we introduce a symmetric monoidal category such that Hom-algebras coincide with algebras in this monoidal category, and similar properties for coalgebras, Hopf algebras and Lie algebras.

💡 Research Summary

The paper “Monoidal Hom‑Hopf algebras” re‑examines the family of Hom‑structures—Hom‑Lie algebras, Hom‑associative algebras, Hom‑coalgebras, and Hom‑Hopf algebras—through the lens of monoidal category theory. The authors begin by isolating the twisting map α, which distinguishes a Hom‑structure from its classical counterpart, and embed it directly into the categorical data. They construct a new symmetric monoidal category 𝓗 whose objects are pairs (A,α) consisting of a vector space (or module) A together with a linear self‑map α, and whose morphisms are α‑linear maps f satisfying f∘α=β∘f. The monoidal product ⊗̂ is defined by (A,α)⊗̂(B,β) = (A⊗B, α⊗β), and the unit object is (k, id). The braiding is the usual flip, now automatically compatible with the twisting maps, making 𝓗 a symmetric monoidal category.

Within 𝓗, an algebra object is precisely a Hom‑algebra (A,μ,α): the multiplication μ:A⊗̂A→A must be α‑linear and satisfy the twisted associativity μ∘(μ⊗̂id)=μ∘(id⊗̂μ)∘a, where a is the associator of 𝓗. Dually, a coalgebra object is a Hom‑coalgebra (A,Δ,ε,α) with α‑linear comultiplication and counit obeying the co‑associativity conditions expressed in the same monoidal language.

The central achievement is the identification of Hom‑Hopf algebras as bialgebra objects equipped with an antipode in 𝓗. The antipode S:A→A must be α‑linear and satisfy the usual Hopf identities μ∘(id⊗̂S)∘Δ = η∘ε = μ∘(S⊗̂id)∘Δ, where η is the unit. Because these identities are formulated using the monoidal structure of 𝓗, they follow automatically from the naturality of the associator and braiding, providing a concise categorical proof of the Hopf axioms in the Hom setting.

The authors also treat Hom‑Lie algebras in the same framework. A bracket