Twisted tensor products of bialgebras and Frobenius algebras

We are interested in when the twisted tensor product of two bialgebras inherits a bialgebra structure in the case where the comultiplication is induced by the inverse of the twisting map. It turns out that this only gives a bialgebra when the twisting map is trivial. However, we find that the twisted tensor product of two Frobenius algebras is always Frobenius with this setup. We also characterize when twisted tensor products of separable algebras are separable, and we prove that twisted tensor products of special Frobenius algebras are special Frobenius. All of the definitions and proofs are given diagrammatically, so these results hold in monoidal categories.

This is joint work with Pablo S. Ocal.