Algebraic structures in comodule categories over weak bialgebras

Abstract

For a bialgebra $L$ coacting on a $\mathbb{k}$-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 Böhm-Caenepeel-Janssen, but the algebras in ${\mathcal{M}}^H$ are not canonically $\mathbb{k}$-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 $\mathbb{k}$-coalgebra and on a Frobenius $\mathbb{k}$-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.

v2: 34 pages. Final version

References in corpus (3)

• Cardy algebras and sewing constraints, I
• Categorification and correlation functions in conformal field theory
• Universal quantum semigroupoids

Cited by in corpus (1)

• Symmetries of algebras captured by actions of weak Hopf algebras