Sweedler Duality for BiHom-associative Algebras

TL;DR

This paper introduces a finite dual construction for BiHom-associative algebras, creating a BiHom-coalgebra structure from an algebra by using linear functionals that annihilate finite-codimensional ideals.

Contribution

It develops a Sweedler-type dual for BiHom-associative algebras, extending duality concepts to infinite-dimensional cases and connecting algebra morphisms to coalgebra morphisms.

Findings

Defines the Sweedler dual for BiHom-algebras.

Proves the dual carries a natural BiHom-coalgebra structure.

Extends the construction to right BiHom-modules and comodules.

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.