On Frobenius and separable algebra extensions in monoidal categories. Applications to wreaths

TL;DR

This paper characterizes Frobenius and separable algebra extensions in monoidal categories, linking properties of extensions to the Frobenius and separability of associated algebras, with applications to wreaths.

Contribution

It provides new criteria for Frobenius and separable extensions in monoidal categories, connecting extension properties to algebra properties within bimodule categories.

Findings

Extension is Frobenius iff $S$ is Frobenius in bimodule category under certain conditions.

Extension is separable iff $S$ is separable algebra when $R$ is separable.

Extension is Frobenius iff $S$ is Frobenius and Nakayama automorphisms align when $R$ is Frobenius and separable.

Abstract

We characterize Frobenius and separable monoidal algebra extensions $i: R\ra S$ in terms given by $R$ and $S$. For instance, under some conditions, we show that the extension is Frobenius, respectively separable, if and only if $S$ is a Frobenius, respectively separable, algebra in the category of bimodules over $R$. In the case when $R$ is separable we show that the extension is separable if and only if $S$ is a separable algebra. Similarly, in the case when $R$ is Frobenius and separable in a sovereign monoidal category we show that the extension is Frobenius if and only if $S$ is a Frobenius algebra and the restriction at $R$ of its Nakayama automorphism is equal to the Nakayama automorphism of $R$. As applications, we obtain several characterizations for an algebra extension associated to a wreath to be Frobenius, respectively separable.