Antisymmetric Infinitesimal Admissible Hom-bialgebras

This paper develops a novel framework for antisymmetric infinitesimal hom-bialgebras by introducing admissible bimodules of hom associative algebras, generalizing the foundational work of Hounkonnou et al. (2019) . We first construct admissible hom-associative bialgebras and rigorously establish their equivalence to matched pairs and Manin triples via a canonical bilinear form. Our approach systematically extends classical bialgebra theory to the hom-associative setting by imposing admissibility conditions on bimodules, which preserve the twisted multiplicativity of hom-structures. Key results include: (1) the characterization of admissible hom associative algebras through their regular bimodules, (2) the derivation of antisymmetric infinitesimal admissible hom-bialgebras as dual structures, and (3) the proof that these bialgebras are categorically equivalent to Manin triples when equipped with a standard invariant bilinear form on the direct sum A ⊕ A∗. This work unifies and generalizes prior studies on hom-Lie bialgebras and the hom Yang-Baxter equation, while providing new tools for deformation theory and non-commutative geometry. The constructions are contextualized within prominent examples, including q-deformations and σ-derivations, bridging connections to the Witt and Virasoro algebras. Our results demonstrate that the hom-associative admissible framework offers a robust algebraic foundation for studying twisted bialgebraic structures.