# Frobenius bimodules and flat-dominant dimensions

## TL;DR

This paper explores the relationships between Frobenius bimodules, Frobenius extensions, and flat-dominant dimensions of algebras, providing new insights into algebraic structures and their invariants, with applications to various algebra classes and Lie algebra enveloping algebras.

## Contribution

It establishes that Frobenius parts of Frobenius extensions are Frobenius, and relates flat-dominant dimensions of algebras linked by Frobenius bimodules, advancing understanding of algebra invariants.

## Key findings

- Frobenius parts of Frobenius extensions are Frobenius extensions.
- Flat-dominant dimensions satisfy inequalities under Frobenius bimodules.
- Universal enveloping algebras of semisimple Lie algebras are QF-3 rings.

## Abstract

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 $\dm(_AX)$ stand for the dominant dimension of an $A$-module $X$. If $_BM_A$ is a Frobenius bimodule, then $\dm(A)\le \dm(_BM)$ and $\dm(B)\le \dm(_A\Hom_B(M, B))$. In particular, if $B\subseteq A$ is a left-split (or right-split) Frobenius extension, then $\dm(A)=\dm(B)$. 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.