# On Frobenius and separable algebra extensions in monoidal categories: applications to wreaths

### Daniel Bulacu

University of Bucharest, Romania### Blas Torrecillas

Universidad de Almería, Almeria, Spain

## Abstract

We characterize Frobenius and separable monoidal algebra extensions $i:R→S$ 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.

## Cite this article

Daniel Bulacu, Blas Torrecillas, On Frobenius and separable algebra extensions in monoidal categories: applications to wreaths. J. Noncommut. Geom. 9 (2015), no. 3, pp. 707–774

DOI 10.4171/JNCG/206