Sweedler Duality for BiHom-associative Algebras
Jiacheng Sun
TL;DR
This work addresses the failure of ordinary linear duality to yield a coalgebra structure for infinite-dimensional BiHom-algebras by introducing a Sweedler-type finite dual $G^{\circ}$. It proves that $G^{\circ}$ is naturally a BiHom-coalgebra with comultiplication $\Delta=\mu^{*}|_{G^{\circ}}$ and twisted maps $\alpha^{\circ}=\alpha^{*}|_{G^{\circ}}$, $\beta^{\circ}=\beta^{*}|_{G^{\circ}}$, and that BiHom-algebra morphisms dualize to BiHom-coalgebra morphisms; the construction is functorial. The duality extends to right BiHom-modules, giving a right BiHom-comodule structure on $M^{\circ}$ under the surjectivity of $\beta$, with $M^{\circ}$ defined by annihilation of finite-codimensional submodules, and morphisms dualize accordingly. Special cases recover the Hom setting ($\alpha=\beta$) and the classical finite dual ($\alpha=\beta=\mathrm{Id}$), highlighting the framework's unification of dualities across BiHom, Hom, and classical contexts.
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 $(G,μ,α,β)$ 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 $μ^{*}$, 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 $β$. The Hom and classical cases are recovered by the specializations $α=β$ and $α=β=\mathrm{Id}$, respectively.