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