# Journal of Noncommutative Geometry

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**Volume 9, Issue 3, 2015, pp. 707–774**

**DOI: 10.4171/JNCG/206**

Published online: 2015-10-29

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

Daniel Bulacu^{[1]}and Blas Torrecillas

^{[2]}(1) University of Bucharest, Romania

(2) Universidad de AlmerÃa, Almeria, Spain

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

*Keywords: *Monoidal category, 2-category, Frobenius algebra, separable algebra, Nakayama automorphism, wreath product

Bulacu Daniel, Torrecillas Blas: On Frobenius and separable algebra extensions in monoidal categories: applications to wreaths. *J. Noncommut. Geom.* 9 (2015), 707-774. doi: 10.4171/JNCG/206