Frobenius bimodules and flat-dominant dimensions

We establish relations between Frobenius parts and between flat-dominant dimensions of algebras linked by Frobenius bimodules. This is motivated by the Nakayama conjecture and an approach of Martinez-Villa to the Auslander-Reiten conjecture on stable equivalences. We show that the Frobenius parts of Frobenius extensions are again Frobenius extensions. Further, let $A$ and $B$ be finite-dimensional algebras over a field $k$, and let $\text{dm}{(}_{A}X)$ stand for the dominant dimension of an $A$-module $X$. If ${}_B{M}_A$ is a Frobenius bimodule, then $\text{dm}(A)\le \text{dm}{(}_{B}M)$ and $\text{dm}(B)\le \text{dm}{(}_A{\text{Hom}}_B(M,B))$. In particular, if $B\subseteq A$ is a left-split (or right-split) Frobenius extension, then $\text{dm}(A)=\text{dm}(B)$. These results are applied to calculate flat-dominant dimensions of a number of algebras: shew group algebras, stably equivalent algebras, trivial extensions and Markov extensions. Finally, we prove that the universal (quantised) enveloping algebras of semisimple Lie algebras are $QF$-$3$ rings in the sense of Morita.