Coquasi-bialgebras with preantipode and rigid monoidal categories

TL;DR

This paper proves that rigid monoidal categories correspond to coquasi-bialgebras with a preantipode, extending the reconstruction theorem and characterizing coquasi-Hopf algebras over fields.

Contribution

It establishes that rigidity in monoidal categories implies the associated coquasi-bialgebra has a preantipode, generalizing Ulbrich's theorem to coquasi-bialgebras.

Findings

Rigid categories lead to coquasi-bialgebras with preantipodes.

Characterization of coquasi-Hopf algebras via rigidity over fields.

Extension of the reconstruction theorem to coquasi-bialgebras.

Abstract

By a theorem of Majid, every monoidal category with a neutral quasi-monoidal functor to finitely-generated and projective $\Bbbk$-modules gives rise to a coquasi-bialgebra. We prove that if the category is also rigid, then the associated coquasi-bialgebra admits a preantipode, providing in this way an analogue for coquasi-bialgebras of Ulbrich's reconstruction theorem for Hopf algebras. When $\Bbbk$ is field, this allows us to characterize coquasi-Hopf algebras as well in terms of rigidity of finite-dimensional corepresentations.