Frobenius monoidal functors from ambiadjunctions and their lifts to Drinfeld centers

TL;DR

This paper characterizes when functors that are both left and right adjoints to strong monoidal functors are Frobenius monoidal, and extends these to braided Frobenius monoidal functors on Drinfeld centers, with applications to Hopf algebra morphisms.

Contribution

It provides new conditions for Frobenius monoidal functors from ambiadjunctions and extends these to braided contexts, with concrete examples from Hopf algebra morphisms.

Findings

Identified conditions using projection formula morphisms for Frobenius monoidal functors.

Extended adjoint functors to braided Frobenius monoidal functors on Drinfeld centers.

Constructed examples from Hopf algebra morphisms via induction.

Abstract

We identify general conditions, formulated using the projection formula morphisms, for a functor that is simultaneously left and right adjoint to a strong monoidal functor to be a Frobenius monoidal functor. Moreover, we identify stronger conditions for the adjoint functor to extend to a braided Frobenius monoidal functor on Drinfeld centers building on our previous work in [arXiv:2402.10094]. As an application, we construct concrete examples of (braided) Frobenius monoidal functors obtained from morphisms of Hopf algebras via induction.