Algebraic structures in comodule categories over weak bialgebras

TL;DR

This paper extends the classical correspondence between comodule algebras and algebra objects in monoidal categories to the setting of weak bialgebras, establishing new categorical isomorphisms and exploring applications in quantum symmetries.

Contribution

It generalizes the categorical characterization of comodule algebras from bialgebras to weak bialgebras, including new isomorphisms and examples involving quantum symmetries.

Findings

Established an isomorphism between categories of right $H$-comodule algebras and algebras in $rac{H}$

Introduced and proved analogous results for $H$ coacting on $$-coalgebras and Frobenius algebras

Constructed a monoidal functor creating weak quantum symmetries from classical quantum symmetries.

Abstract

For a bialgebra $L$ coacting on a $\Bbbk$-algebra $A$, a classical result states that $A$ is a right $L$-comodule algebra if and only if $A$ is an algebra in the monoidal category $\mathcal{M}^{L}$ of right $L$-comodules; the former notion is formulaic while the latter is categorical. We generalize this result to the setting of weak bialgebras $H$. The category $\mathcal{M}^H$ admits a monoidal structure by work of Nill and B\"{o}hm-Caenepeel-Janssen, but the algebras in $\mathcal{M}^H$ are not canonically $\Bbbk$-algebras. Nevertheless, we prove that there is an isomorphism between the category of right $H$-comodule algebras and the category of algebras in $\mathcal{M}^H$. We also recall and introduce the formulaic notion of $H$ coacting on a $\Bbbk$-coalgebra and on a Frobenius $\Bbbk$-algebra, respectively, and prove analogous category isomorphism results. Our work is inspired by the physical applications of Frobenius algebras in tensor categories and by symmetries of algebras with a base algebra larger than the ground field (e.g. path algebras). We produce examples of the latter by constructing a monoidal functor from a certain corepresentation category of a bialgebra $L$ to the corepresentation category of a weak bialgebra built from $L$ (a "quantum transformation groupoid"), thereby creating weak quantum symmetries from ordinary quantum symmetries.