Symmetric algebras of corepresentations and smash products

TL;DR

This paper explores the properties of Frobenius and symmetric algebras within the category of right comodules over a Hopf algebra, focusing on the relationship between comodule algebras and their smash products.

Contribution

It establishes new connections between Frobenius and symmetric properties of comodule algebras and their smash products over finite-dimensional Hopf algebras.

Findings

Identifies conditions under which comodule algebras are Frobenius or symmetric

Analyzes the impact of the cosovereign property on symmetry

Links properties of $A$ and $A\# H^*$ in the Hopf algebra context

Abstract

We investigate Frobenius algebras and symmetric algebras in the monoidal category of right comodules over a Hopf algebra $H$; for the symmetric property $H$ is assumed to be cosovereign. If $H$ is finite dimensional and $A$ is an $H$-comodule algebra, we uncover the connection between $A$ and the smash product $A\# H^*$ with respect to the Frobenius and symmetric properties.