Sweedler Duality for BiHom-associative Algebras

Abstract

Motivated by the fact that ordinary linear duality does not in general produce a coalgebra structure from an infinite-dimensional algebra, we develop a Sweedler-type finite dual construction for BiHom-associative algebras. For a BiHom-algebra $(G,\mu,\alpha,\beta)$ over a field, we define its Sweedler dual $G^{\circ}\subseteq G^{*}$ as the subspace of linear functionals annihilating a finite-codimensional BiHom-ideal of $G$. We prove that $G^{\circ}$ carries a natural BiHom-coalgebra structure whose comultiplication is the restriction of $\mu^{*}$, and that BiHom-algebra morphisms induce BiHom-coalgebra morphisms on Sweedler duals. We further extend this construction to right BiHom-modules, obtaining right BiHom-comodules over $G^{\circ}$ under a surjectivity assumption on the twisting map $\beta$. The Hom and classical cases are recovered by the specializations $\alpha=\beta$ and $\alpha=\beta=\mathrm{Id}$, respectively.